Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza

1. Introduction

Spectral  sections of  families of  self-adjoint  Fredholm  operators were introduced  by  Melrose  and  Piazza  [$MP_{\hskip 0.35002pt1}$]  for  the needs of  index  theory.   The basic result  about  spectral  sections  is  a  theorem of  Melrose and  Piazza  to  the effect  that  a family admits a spectral  section  if  and  only  if  its analytic  index  vanishes.   See  [$MP_{\hskip 0.35002pt1}$],  Proposition  1.   Melrose  and  Piazza  [$MP_{\hskip 0.35002pt1}$]  gloss over  the definition of  the analytic  index and  the notion of  a  trivialization of  a  Hilbert  bundle  implicitly  used  in  their  proof.   Since passing  from one  trivializations of  a  Hilbert  bundle  to another rarely  preserves  the norm continuity of  families of  Hilbert  space operators,   the straightforward  interpretation of  Melrose–Piazza  proof  works only  for  families of  operators in a fixed  Hilbert  space.   The author  learned about  this  issue  from  M.  Prokhorova  [$P_{\hskip 0.35002pt1}$].

Recently  the author  proved  a  general  version of  this  theorem of  Melrose–Piazza  as a byproduct  of  a  theory developed  in  [$I_{\hskip 0.70004pt1}$],  [$I_{\hskip 0.70004pt2}$].   See  [$I_{\hskip 0.70004pt2}$],   Corollary  6.2.   After  learning about  Melrose  approach  [M]  to clarifying  [$MP_{\hskip 0.35002pt1}$],   the author  realized  that  a  less general  version of  Melrose–Piazza  theorem can be disentangled  from  the  theory of  [$I_{\hskip 0.70004pt1}$],  [$I_{\hskip 0.70004pt2}$].   This version  is  still  more general  than  the original  one  [$MP_{\hskip 0.35002pt1}$].   An analysis of  the resulting  proof  led  to  a fairly simple way  to prove  the original  Melrose–Piazza  theorem,   or,   rather,   its  “axiomatic”  version.

The present  paper  is  devoted  to  the proofs  of  these  two versions of  Melrose–Piazza  theorem.   The ideas of  Atiyah–Singer  [AS]  play a  key  role,   but,   in contrast  with  [M]  and  [AS],   compact  operators are not  used even  implicitly  in  the proof  of  the first,   more general,   version.   The proof  of  the second version  is  closer  to  the ideas of  Melrose  [M]  and  uses compact  operators and  some ideas of  Atiyah–Segal  [ASe].   Both  proofs are presented as complements  to  [$MP_{\hskip 0.35002pt1}$].   This forces us  to  deal only with compact  spaces of  parameters.   If  one replaces  references  to  [$MP_{\hskip 0.35002pt1}$]  by  references  to  [$I_{\hskip 0.70004pt2}$],   the same proofs would  work for paracompact  spaces.

Both  proofs depend on  a  theorem about  the existence of  trivializations appropriately adapted  to families.   See  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  below  for  the statement  and  [$I_{\hskip 0.70004pt2}$],   Theorem  4.6,   for a proof.   The case of  triangulable bases  is  considered  in  [$I_{\hskip 0.70004pt2}$],   Theorem  4.5,   and  is  independent  from  the rest  of  [$I_{\hskip 0.70004pt2}$].   Also,   we refer  to  [$I_{\hskip 0.70004pt2}$]  for  a detailed  proof  of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza,   which  is  treated  in  [$MP_{\hskip 0.35002pt1}$]  as obvious.   It  is  also independent  from  the rest  of  [$I_{\hskip 0.70004pt2}$].

In  [$MP_{\hskip 0.35002pt2}$]  Melrose  and  Piazza  proved an odd $\mathbf{Z}_{\hskip 0.70004pt2}$-graded version of  their  theorem.   There  is  no doubt  that  the methods of  the present  paper  work  in  this case also.   Cf.  [$I_{\hskip 0.70004pt1}$],   [$I_{\hskip 0.70004pt2}$].

2. Basic  definitions

The framework.   Let  $H$  be a separable infinite dimensional  Hilbert  space over  $\mathbf{C}$.   Let  $\mathbb{H}$  be a  locally  trivial  Hilbert  bundle with  a paracompact  base  $X$  and  fibers isomorphic  to  $H$.   We will  treat  $\mathbb{H}$  as a family  $H_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  of  Hilbert  spaces.   We will  assume  that  the space  $X$  is  paracompact.   Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  be a family of  self-adjoint  Fredholm  operators.   We will  assume  that  either  all  operators $A_{\hskip 0.70004ptx}$ are bounded,   or  that  they are closed and densely defined.   In  the first  case we will  assume  that  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  continuous as a self-map of  the  total  space of $\mathbb{H}$.   In  the second case we will  assume  that  the  bounded  transform  $\gamma\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$,   where  $\gamma\hskip 1.49994pt(\hskip 1.49994ptt\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptt\hskip 1.49994pt(\hskip 1.49994pt1\hskip 1.99997pt+\hskip 1.99997ptt^{\hskip 0.70004pt2}\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1/2}$,   has  this property.   By  a well  known  reason we assume  that  operators  $A_{\hskip 0.70004ptx}$  are neither essentially  positive,   nor  essentially negative.

Enhanced operators.   An  enhanced  (self-adjoint  Fredholm)  operator  is  a pair $(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.00006pt)$,  where  $A\hskip 1.00006pt\colon\hskip 1.00006ptH\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$  is  a self-adjoint  Fredholm  operator,  $\varepsilon\hskip 1.99997pt>\hskip 1.99997pt0$,   such  that  $\varepsilon\hskip 0.50003pt,\hskip 1.99997pt-\hskip 1.99997pt\varepsilon\hskip 1.99997pt\not\in\hskip 1.99997pt\sigma\hskip 1.49994pt(\hskip 1.49994ptA\hskip 1.49994pt)$  and  the interval  $[\hskip 1.49994pt-\hskip 1.99997pt\varepsilon\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt]$  is  disjoint  from  the essential  spectrum of  $A$.   Then  the spectral  projection  $P_{\hskip 0.70004pt[\hskip 0.70004pt-\hskip 0.70004pt\varepsilon\hskip 0.35002pt,\hskip 1.39998pt\varepsilon\hskip 0.70004pt]}\hskip 1.49994pt(\hskip 1.49994ptA\hskip 1.49994pt)$  has finitely dimensional  image.   If  $(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced operator  and  $A^{\prime}\hskip 1.00006pt\colon\hskip 1.00006ptH\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$  is  a self-adjoint  Fredholm  operator sufficiently close  to  $A$  in  the norm  topology or  in  the uniform  resolvent  sense,   then  $(\hskip 1.49994ptA^{\prime}\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  also an enhanced operator.   The spectral  projection  $P_{\hskip 0.70004pt[\hskip 0.70004pt-\hskip 0.70004pt\varepsilon\hskip 0.35002pt,\hskip 1.39998pt\varepsilon\hskip 0.70004pt]}\hskip 1.49994pt(\hskip 1.49994ptA^{\prime}\hskip 1.49994pt)$ continuously depend on  $A^{\prime}$ for  $A^{\prime}$ sufficiently close  to  $A$.

Fredholm  families.   The family  $A_{\hskip 0.70004ptx}\hskip 0.50003pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  said  to be a  Fredholm  family  if  all  operators $A_{\hskip 0.70004ptx}$ are  Fredholm  and  for every  $z\hskip 1.99997pt\in\hskip 1.99997ptX$  there exists  $\varepsilon\hskip 3.99994pt=\hskip 3.99994pt\varepsilon_{\hskip 0.70004ptz}\hskip 1.99997pt>\hskip 1.99997pt0$  and a neighborhood  $U_{\hskip 0.70004ptz}$  of  $z$ such  that  $(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.00006pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced  operator  for every $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.70004ptz}$,   the subspaces

continuously depend on  $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.70004ptz}$,   and  the operators  $V_{\hskip 0.35002pty}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptV_{\hskip 0.35002pty}$ induced  by  $A_{\hskip 0.70004pty}$  also continuously depend on  $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.70004ptz}$.   Two  families  $A_{\hskip 0.70004ptx}\hskip 0.50003pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  and  $A^{\prime}_{\hskip 0.70004ptx}\hskip 0.50003pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  are said  to be  Fredholm  homotopic  if  there exists  a  homotopy  $A_{\hskip 1.04996ptx\hskip 0.35002pt,\hskip 0.70004ptu}$,  $(\hskip 1.49994ptx\hskip 0.50003pt,\hskip 1.99997ptu\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997ptX\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$  which  is  Fredholm  as a family and  such  that  $A_{\hskip 1.04996ptx\hskip 0.35002pt,\hskip 0.70004pt0}\hskip 3.99994pt=\hskip 3.99994ptA_{\hskip 0.70004ptx}$  and  $A_{\hskip 1.04996ptx\hskip 0.35002pt,\hskip 0.70004pt1}\hskip 3.99994pt=\hskip 3.99994ptB_{\hskip 0.70004ptx}$  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$.

Strictly  Fredholm  families.   Suppose  first  that  $\mathbb{H}$  is  the  trivial  bundle with  the fiber  $H$.   We will  say  that  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  strictly  Fredholm  if  it  is  Fredholm  and  for every  $z\hskip 1.99997pt\in\hskip 1.99997ptX$  there exists  $\varepsilon\hskip 1.99997pt>\hskip 1.99997pt0$  and  a neighborhood  $U_{\hskip 0.35002ptz}$  of  $z$  such  that  $(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced operator for every  $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$ and  the spectral  projection  $P_{\hskip 0.70004pt[\hskip 1.04996pt\varepsilon\hskip 0.35002pt,\hskip 1.39998pt\infty\hskip 0.70004pt)}\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)$  continuously depends on  $y$  in  the norm  topology  for  $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$.

In  general,  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  said  to be a  strictly  Fredholm  family  if  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  there exists  a neighborhood  $U_{\hskip 0.35002ptz}$  of  $z$  and a  local  trivialization of  $\mathbb{H}$  over  $U_{\hskip 0.35002ptz}$  turning  the restriction  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a strictly  Fredholm  family  in  the above sense.   Such a  local  trivialization  is  said  to be  strictly adapted  to  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   A  trivialization of  $\mathbb{H}$  is  said  to be  strictly adapted  to  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  if  for every  $z\hskip 1.99997pt\in\hskip 1.99997ptX$  there  exists  a neighborhood  $U_{\hskip 0.35002ptz}$  of  $z$  such  that  its restriction  to  $U_{\hskip 0.35002ptz}$  is  strictly adapted  to  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.

Fully  Fredholm  families.   Suppose  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a family of  bounded operators.   We  say  that  it  is  fully  Fredholm  if  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  there exists  a neighborhood  $U_{\hskip 0.35002ptz}$  of  $z$  and a  local  trivialization of  $\mathbb{H}$  over  $U_{\hskip 0.35002ptz}$  turning  the restriction  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a family continuous in  the norm  topology.   Such a  local  trivialization  is  said  to be  fully adapted  to  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   Clearly,   a  fully  Fredholm  family  is  strictly  Fredholm.

If  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a family of  closed densely defined operators,   we will  say  that  it  is  fully  Fredholm  if  the bounded  transform  $\gamma\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  is  fully  Fredholm.   The  fully adapted  ( local )  trivializations of  such families are defined  in  the obvious way.

Polarizations and  restricted  Grassmannians.   A  polarization  of  a  Hilbert  space  $K$  is  a presentation of  $K$  as an orthogonal  direct  sum  $K\hskip 3.99994pt=\hskip 3.99994ptK_{\hskip 0.70004pt-}\hskip 1.00006pt\oplus\hskip 1.00006ptK_{\hskip 0.70004pt+}$  of  two closed  infinitely dimensional  subspaces  $K_{\hskip 0.70004pt-}\hskip 1.00006pt,\hskip 3.99994ptK_{\hskip 0.70004pt+}$.   A polarization  leads  to  the  restricted  Grassmannian  $\operatorname{G{\hskip 0.50003pt}r}$,   the space of  subspaces  $L\hskip 1.99997pt\subset\hskip 1.99997ptK$  commensurable  with $K_{\hskip 0.70004pt-}$,   i.e.  such  that  the intersection  $L\hskip 1.00006pt\cap\hskip 1.00006ptK_{\hskip 0.70004pt-}$  is  closed and  has finite codimension  in  both  $L$  and  $K_{\hskip 0.70004pt-}$.   The  topology of  $\operatorname{G{\hskip 0.50003pt}r}$  is  defined  by  the norm  topology of  orthogonal  projections  $K\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL$.

Grassmannian  bundles and  weak  spectral  sections.   Suppose  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.49994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a strictly  Fredholm  family.   If  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  and  $(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced operator,   then

is  a polarization of  $H_{\hskip 0.70004ptx}$.   This polarization  leads  to  a  restricted  Grassmannian,   which  we will  denote by  $\operatorname{G{\hskip 0.50003pt}r}\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)$.   Since  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  strictly  Fredholm,   the family  of  restricted  Grassmannians  $\operatorname{G{\hskip 0.50003pt}r}\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  forms a  locally  trivial  bundle  $\bm{\pi}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt\operatorname{G{\hskip 0.50003pt}r}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$.   Continuous sections of  this bundle are called  weak spectral  sections  of  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.

Discrete-spectrum  families and spectral  sections.   The family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  said  to be a  discrete-spectrum  family  if  for every  $\lambda\hskip 1.99997pt\in\hskip 1.99997pt\mathbf{R}$  the family  of  operators  $A_{\hskip 0.70004ptx}\hskip 1.99997pt-\hskip 3.00003pt\lambda$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a  Fredholm  family.   In  particular,   operators  $A_{\hskip 0.70004ptx}\hskip 1.99997pt-\hskip 3.00003pt\lambda$  are  Fredholm  for every  $\lambda\hskip 1.99997pt\in\hskip 1.99997pt\mathbf{R}$.   The operators $A_{\hskip 0.70004ptx}$ in  such a family  have discrete spectrum and cannot  be bounded.   Since  they are self-adjoint,   they are necessarily closed and densely defined.

Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  be a strictly  Fredholm  and  discrete-spectrum  family.   A  weak spectral  section  $S\hskip 1.00006pt\colon\hskip 1.00006ptX\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\operatorname{G{\hskip 0.50003pt}r}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}\hskip 1.49994pt)$  is  said  to be a  spectral  section  if

for a continuous function $r\hskip 1.00006pt\colon\hskip 1.00006ptX\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{R}_{\hskip 1.04996pt>\hskip 0.70004pt0}$  and every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$.   This  is  a generalization of  the notion of  spectral  sections introduced  by  Melrose  and  Piazza  [$MP_{\hskip 0.35002pt1}$].

Discrete-spectrum  families and classical  operator  topologies.   The material  of  this subsection  is  not  used  in  the rest  of  the paper.

The discrete-spectrum  families were introduced  by  the author  [$I_{\hskip 0.70004pt2}$]  as a natural  analogue of  the notion of  a  Fredholm  family  for families of  operators with discrete spectrum.   At  the same  time  such families are exactly  the families for which  the proof  of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  below about  the existence of  spectral  sections works.

Recently  M.  Prokhorova  [$P_{\hskip 0.35002pt3}$]  related  the notion of  a discrete-spectrum  family with  classical  continuity  properties.   Note  that  every operator in a discrete-spectrum  family  is  a closed densely defined operator with compact  resolvent.   Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  be a family  of  self-adjoint  operators with compact  resolvent  in a fixed  Hilbert  space  $H$.   Then  $A_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a discrete-spectrum  family  if  and  only  if  $A_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  continuous in  the  topology of  convergence in  the norm  resolvent  sense.   Also,   for discrete-spectrum  families every  strictly adapted  local  trivialization  is  fully adapted.   In  particular,   a discrete-spectrum  family  is  strictly  Fredholm  if  and  only  if  it  is  fully  Fredholm.   See  [$P_{\hskip 0.35002pt3}$],   Theorems  2  and  3.

3. Basic  theorems

Classical  contractibility  theorems.   Let  $K$  be a separable infinite dimensional  Hilbert  space.   Let  us consider  the group of  isometries  $K\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  and denote by $U\hskip 1.00006pt(\hskip 1.49994ptK\hskip 1.49994pt)$ this group equipped with  the norm  topology.   The group $U\hskip 1.00006pt(\hskip 1.49994ptK\hskip 1.49994pt)$  is  contractible by a  theorem of  Kuiper  [K].

Another useful  topology  on  this group  is  the compact-open one.   We will  denote by $\mathcal{U}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 1.49994pt)$  this group equipped  with  topology induced  from  the product  of  compact-open  topologies by  the map  $g\hskip 3.99994pt\longmapsto\hskip 3.99994pt(\hskip 1.49994ptg\hskip 0.50003pt,\hskip 1.99997ptg^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt)$.   Actually,   this  topology coincides with  the strong operator  topology,   but  the author prefers  to ignore  this fact.   The group $\mathcal{U}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 1.49994pt)$  is  contractible by a  theorem of  Atiyah  and  Segal  [ASe],   who adapted an argument  of  Dixmier  and  Douady  [DD].

Another  important  space  is  the space of  polarizations  $H\hskip 3.99994pt=\hskip 3.99994ptH_{\hskip 0.70004pt-}\hskip 1.00006pt\oplus\hskip 1.00006ptH_{\hskip 0.70004pt+}$  with  the  topology defined  by  the norm  topology of  orthogonal  projections  $H\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004pt-}$.   By an observation of  Atiyah  and  Singer  [AS],   Kuiper’s  theorem  implies  that  this space  is  also contractible.

3.1. Theorem.   If  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a strictly  Fredholm  family,   then  there exists a  trivialization of  $\mathbb{H}$  strictly  adapted  to  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   

Proof.   The proof  is  simpler  if  there exists a  triangulation of  the space  $X$,   perhaps infinite.   In  this case one can  argue by  an  induction  by skeletons,   using  at  each  step both  the contractibility of  the space of  polarizations and  the contractibility of  the groups $\mathcal{U}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 1.49994pt)$.   The  latter  is  applied  to  $K\hskip 3.99994pt=\hskip 3.99994ptH_{\hskip 0.70004pt-}\hskip 1.00006pt,\hskip 3.00003ptH_{\hskip 0.70004pt+}$  for polarizations  $H\hskip 3.99994pt=\hskip 3.99994ptH_{\hskip 0.70004pt-}\hskip 1.00006pt\oplus\hskip 1.00006ptH_{\hskip 0.70004pt+}$.   The case of  triangulable space  $X$  is  sufficient  for applications.   The general  case of  a paracompact  space  $X$  requires more sophisticated  tools from  the homotopy  theory.   See  [$I_{\hskip 0.70004pt2}$],   Theorems  4.5  and  4.6.    $\blacksquare$

3.2. Theorem.   If  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a  discrete-spectrum  and  strictly  Fredholm  family,   then every  weak  spectral  section  of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  homotopic  to a spectral  section.   

Proof.   This  is  an explicit  form of  the  last  paragraph  in  the proof  of  Proposition  1  of  Melrose  and  Piazza  [$MP_{\hskip 0.35002pt1}$],   who claim  that  a weak spectral  section can  be  transformed  into a spectral  section  “simply  by  smoothly  truncating  the eigenfunction  expansion”.   See  [$I_{\hskip 0.70004pt2}$],   Theorem  6.1  for a detailed  geometric  version of  this argument.    $\blacksquare$

3.3. Theorem.   Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  be a norm continuous family of  Fredholm  self-adjoint  operators in a fixed  Hilbert  space  $H$.   Then  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  admits a weak  spectral  section  if  and  only  if  it  is  homotopic  in  the class of  such  families  to a family  of  invertible operators.   

Proof.   For compact  $X$  the proof  is  contained  in  [$MP_{\hskip 0.35002pt1}$].   See  [$MP_{\hskip 0.35002pt1}$],   the proof  of  Proposition  1.   For  general  paracompact  $X$  this follows from  [$I_{\hskip 0.70004pt2}$].   We omit  the details.    $\blacksquare$

Remark.   Suppose  that  $A_{\hskip 0.70004ptx}\hskip 0.50003pt\colon\hskip 0.50003ptH\hskip 1.00006pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.00006ptH$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a  family of  (closed densely defined )  self-adjoint  operators.   The results of  Prokhorova  [$P_{\hskip 0.35002pt3}$]  mentioned at  the end of  Section  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  imply  that  $A_{\hskip 0.70004ptx}$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a discrete-spectrum and strictly  Fredholm  family  if  and only  if  $A_{\hskip 0.70004ptx}$  are operators with compact  resolvent  and  the family of  bounded  transforms  $\gamma\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 1.49994pt)$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  is  norm  continuous.   In view of  this equivalence,   the analogues of  Theorems  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  for such families with paracompact  $X$  follow  from  the results of  Prokhorova  [$P_{\hskip 0.35002pt2}$].   See  [$P_{\hskip 0.35002pt2}$],   Theorem  4.4  (note  that  every weak spectral  section  is  a generalized spectral  section).

4. The  first  proof

Finite-polarized  replacements.   We will  call  a self-adjoint  operator  $A\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  in a  Hilbert  space  $K$  finite-polarized   if  $\|\hskip 1.99997ptA\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt1$,   the essential  spectrum of  $A$  consists of  two points  $-\hskip 1.99997pt1\hskip 0.50003pt,\hskip 1.99997pt1$,   and  the spectral  projection  $P_{\hskip 1.04996pt(\hskip 0.70004pt-\hskip 1.39998pt1\hskip 0.35002pt,\hskip 1.39998pt1\hskip 1.04996pt)}\hskip 1.49994pt(\hskip 1.49994ptA\hskip 1.49994pt)$  is  an operator of  finite rank.   If  we omit  the  last  property,   we will  get  exactly  the operators from  the space  $\hat{F}_{\hskip 0.70004pt*}$ from  [AS],   Section  2.

Suppose  that  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  Fredholm.   We would  like  to replace  it  by a family  of  finite-polarized operators  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  without  affecting  analytic  index  and  weak  spectral  sections.   This can  be done by  a spectral  deformation  similar  to one used  in  [AS].   Let  us choose  for each  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  a neighborhood  $U_{\hskip 0.70004ptx}$  of  $x$ and a number  $\varepsilon_{\hskip 0.70004ptx}\hskip 1.99997pt\in\hskip 1.99997pt(\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt)$  such  that  the properties from  the definition of  Fredholm  families hold.   Since  $X$  is  paracompact,   we can assume  that  for some  $\Sigma\hskip 1.99997pt\subset\hskip 1.99997ptX$  the family  $U_{\hskip 0.35002pta}\hskip 1.00006pt,\hskip 1.99997pta\hskip 1.99997pt\in\hskip 1.99997pt\Sigma$  is  a  locally  finite covering of  $X$  and  there exists a partition of  unity  subordinated  to  this covering.   Let  $r_{\hskip 0.35002pta}\hskip 1.00006pt\colon\hskip 1.00006ptX\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{R}_{\hskip 1.39998pt\geqslant\hskip 0.70004pt0}$  be  the function from  this partition of  unity corresponding  to  $a\hskip 1.99997pt\in\hskip 1.99997pt\Sigma$.   Let

Then $r\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997pt(\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt)$ and  $r\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)\hskip 1.99997pt\leqslant\hskip 1.99997pt\max\hskip 1.99997pt\varepsilon_{\hskip 0.70004pta}$,   where  the maximum  is  over  $a$  such  that  $x\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002pta}$,   for every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$.   It  follows  that  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  the essential  spectrum of  $A_{\hskip 0.70004ptx}$  is  disjoint  from  $(\hskip 1.49994pt-\hskip 1.99997ptr\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)\hskip 0.50003pt,\hskip 1.99997ptr\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)\hskip 1.99997pt)$.   For  $r\hskip 1.99997pt>\hskip 1.99997pt0$  let  $\chi_{\hskip 0.70004ptr}\hskip 1.00006pt\colon\hskip 1.00006pt\mathbf{R}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{R}$  be  an odd  increasing  function  such  that

We assume  that  $\chi_{\hskip 0.70004ptr}$ continuously depends on $r$ in  the $\sup$-norm  topology.   For  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  let

The operators  $A^{\prime}_{\hskip 0.70004ptx}$  are finite-polarized and  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a  Fredholm  family.   The family  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  our  finite-polarized  replacement  of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  .

4.1. Lemma.   If  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  strictly  Fredholm,   then  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  also strictly  Fredholm,   and  these  two families have  the same weak spectral  sections.   

Proof.   Clearly,   if  $0\hskip 1.99997pt<\hskip 1.99997pt\varepsilon\hskip 1.99997pt<\hskip 1.99997ptr\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)/2$,   then

It  follows  that  if  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  strictly  Fredholm,   then  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  also strictly  Fredholm.   This also implies  that  the bundles  $\bm{\pi}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt\operatorname{G{\hskip 0.50003pt}r}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$  and  $\bm{\pi}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}^{\prime}\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt\operatorname{G{\hskip 0.50003pt}r}\hskip 1.49994pt(\hskip 1.49994pt\mathbb{A}^{\prime}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$  corresponding  to  these  two families are equal  (not  only  isomorphic,   but  equal ).   Therefore  these bundles have  the same sections,   i.e.  weak  spectral  sections are  the same.    $\blacksquare$

4.2. Lemma.   Every  local  trivialization  strictly adapted  to  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  adapted  to  the family  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   In  particular,   if  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  strictly  Fredholm,   then  the family  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm.   

Proof.   Let  $z\hskip 1.99997pt\in\hskip 1.99997ptX$.   There exist  $\varepsilon\hskip 1.99997pt>\hskip 1.99997pt0$  and a neighborhood  $U_{\hskip 0.70004ptz}$  of  $z$  such  that  $(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.00006pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced operator  and  $\varepsilon\hskip 1.99997pt<\hskip 1.99997ptr\hskip 1.49994pt(\hskip 1.49994pty\hskip 1.49994pt)/2$  for every  $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.70004ptz}$.   If  $U$  is  sufficiently small,   then a strictly adapted  trivialization over  $U$  turns  the family

into a norm-continuous  one.   But  $A^{\prime}_{\hskip 0.70004pty}\hskip 1.00006pt,\hskip 1.99997ptu\hskip 1.99997pt\in\hskip 1.99997ptU$  differs from  it  by  a norm continuous family of  operators of  finite rank.   This implies  that  the same  local  trivialization  turns  the family  $A^{\prime}_{\hskip 0.70004pty}\hskip 1.00006pt,\hskip 1.99997ptu\hskip 1.99997pt\in\hskip 1.99997ptU$  into a norm continuous one.    $\blacksquare$

4.3. Lemma.   If  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm,   then  there exists a  fully  Fredholm  homotopy  between  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  and  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   

Proof.   It  is  sufficient  to consider  the case when  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a family of  bounded operators,   since  the case of  closed densely defined operators reduces  to  it  by applying  the bounded  transform  to both  families.   The  linear homotopies between  the identity  $\operatorname{id}\hskip 1.00006pt\colon\hskip 1.00006pt\mathbf{R}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{R}$  and  the functions  $\chi_{\hskip 0.70004ptr\hskip 1.04996pt(\hskip 1.04996ptx\hskip 1.04996pt)}$  define a homotopy  between  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  and  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   If  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm,   then  this homotopy  is  also fully  Fredholm.    $\blacksquare$

Atiyah–Singer  approach  to  the analytic  index.   Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$,  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  be a  Fredholm  family of  bounded operators in a fixed  Hilbert  space $H$.   Let  us consider  the family  of  Fredholm  operators  (not  assumed  to be self-adjoint )  defined  by  the formula

where  $\operatorname{id}_{\hskip 1.04996ptH}$  is  the identity operator  $H\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$  and  $i\hskip 3.99994pt=\hskip 3.99994pt\sqrt{\hskip 1.00006pt-\hskip 1.99997pt1}$.   The idea  is  to define  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  as  the analytic  index of  the family  $B_{\hskip 0.70004ptx\hskip 0.35002pt,\hskip 1.39998ptt}\hskip 1.00006pt,\hskip 3.99994pt(\hskip 1.49994ptx\hskip 0.50003pt,\hskip 1.99997ptt\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997ptX\hskip 1.00006pt\times\hskip 1.00006pt\ [\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt]$.   The  latter  should  be considered  relatively  to  $X\hskip 1.00006pt\times\hskip 1.00006pt0\hskip 1.99997pt\cup\hskip 1.99997ptX\hskip 1.00006pt\times\hskip 1.00006pt\pi$,   reflecting  the fact  that  $B_{\hskip 0.70004ptx\hskip 0.35002pt,\hskip 1.39998pt0}\hskip 3.99994pt=\hskip 3.99994pt\operatorname{id}_{\hskip 1.04996ptH}$  and  $B_{\hskip 0.70004ptx\hskip 0.35002pt,\hskip 1.39998pt\pi}\hskip 3.99994pt=\hskip 3.99994pt-\hskip 1.99997pt\operatorname{id}_{\hskip 1.04996ptH}$  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptX$.   More precisely,   the analytic  index  is  either an element  of  $K\hskip 1.49994pt(\hskip 1.49994pt\Sigma\hskip 1.00006ptX\hskip 1.49994pt)$,   where  $\Sigma\hskip 1.00006ptX$  is  the suspension of  $X$  or  a homotopy class of  a map  from  $X\hskip 1.00006pt\times\hskip 1.00006pt\ [\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt]$  to an appropriate classifying space with respect  to homotopies fixed on  $X\hskip 1.00006pt\times\hskip 1.00006pt0$  and  $X\hskip 1.00006pt\times\hskip 1.00006pt\pi$.

If  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm,   then  the family  (1)  is  fully  Fredholm  in  the sense  that  over an open neighborhood of  every  $(\hskip 1.49994ptx\hskip 0.50003pt,\hskip 1.99997ptt\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997ptX\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt]$  there  exists a  trivialization of  $\mathbb{H}\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt]$  turning  this family  into a norm-continuous family.   The definition of  the analytic  index  in  the non-self-adjoint  case  [A],   [$AS_{\hskip 0.35002pt4}$]  trivially  extends  to such  families,   at  least  for  compact  $X$.   Hence we can extend  the definition of  the analytic  index  to fully  Fredholm  families of  self-adjoint  operators.   Clearly,   it  is  invariant  under  fully  Fredholm  homotopies.   If  $X$  is  only  paracompact,   one needs  to use  Segal’s  definition  [S]  of  the analytic  index and more  work  is  required.   See  [$I_{\hskip 0.70004pt2}$],   Sections  7  and  8,   for  this case.

It  is  easy  to see  that  the bounded  transform  $\gamma\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  has  the same analytic  index as  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   This allows  to extend  the above definition of  the analytic index  to families of  closed densely defined self-adjoint  Fredholm  operators in  $H$.   Namely,   given such a family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$,   one defines its index as  the index of  the bounded  transform  $\gamma\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.

Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  allows  to extend  this definition  to strictly  Fredholm  families  of  operators in  the fibers of  a  Hilbert  bundle.   The index does not  depend on  the choice of  trivialization  by  the relative version of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza.   See  [$I_{\hskip 0.70004pt2}$],   Theorem  4.7  for  the  latter.

4.4. Theorem   Suppose  that  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm.   A weak  spectral  section  for  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  exists  if  and  only  if  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  vanishes.   

Proof.   We will  limit  ourselves by  the case of  compact  $X$.   Suppose  that  there exists a weak  spectral  section.   Then  the arguments  in  the first  part  of  the proof  of  Proposition  1  in  [$MP_{\hskip 0.35002pt1}$]  together  with  the principle of  uniform  boundedness show  that  there exists a  fully  Fredholm  homotopy  between  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  and  a family  of  invertible operators.   Hence we may assume  that  operators  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  are invertible.   Then  all  operators  (1)  are also invertible.   In  the non-self-adjoint  case  the analytic  index of  families of  invertible operators vanishes essentially  by  the definition.   The  “only  if”  part  follows.

Suppose now  that  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  vanishes.   Lemma  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  implies  that  the analytic  index of  our  finite-polarized  replacement  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  also vanishes.   By  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  there exists a strictly adapted  to  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  trivialization of  $\mathbb{H}$.   By  Lemma  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  such  trivialization  turns  the family  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  into a norm continuous one.   The analytic index of  a norm continuous family vanishes  if  and  only  if  it  is  homotopic  to a constant  family.   Since a self-adjoint  operator can be deformed  to an  invertible self-adjoint  operator,   in  this case  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  homotopic  to a family of  invertible operators.   By  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  this implies  that  there exists a  weak  spectral  section of  $A^{\prime}_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   Lemma  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  implies  that  this weak  spectral  section  is  also a weak  spectral  section  for  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$.   This proves  the  “if”  part.    $\blacksquare$

4.5. Corollary.   Suppose  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a discrete-spectrum  and  fully  Fredholm  family.   A spectral  section  for  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  exists  if  and  only  if  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  vanishes.   

Proof.   It  is  sufficient  to combine  Theorems  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza.    $\blacksquare$

Remark.   In  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Corollary  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  it  is  sufficient  to assume  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a strictly  Fredholm  family.   This follows  from  [$I_{\hskip 0.70004pt2}$],   Theorems  5.2  and  Corollary  6.2,   if  one  takes into account  the equivalence of  the definition of  the analytic index  suggested  in  [$I_{\hskip 0.70004pt2}$]  and  the classical  definition.   This equivalence  is  proved  in  [$I_{\hskip 0.70004pt2}$],   Theorem  8.5.   The proofs automatically  work  for paracompact  $X$.   The proof  of  the equivalence of  two definitions of  the analytic  index  is  simpler  for compact  $X$,   where  in  the non-self-adjoint  case one can use  Atiyah’s  definition  [A].   For  paracompact  $X$  one needs  to use  Segal’s  definition  [S].   The resulting proof  of  the  “only  if”  parts depends on  the  theory developed  in  [$I_{\hskip 0.70004pt1}$].

Families of  elliptic operators.   Let $\mathbb{M}$ be a  locally  trivial  bundle over $X$ with closed  manifolds as fibers.   Let  us  consider a continuous  family  of  elliptic pseudo-differential  operators of  order  $0$  acting on  fibers of  $\mathbb{M}$.   It  defines a family of  bounded operators  acting in  fibers of  a  Hilbert  bundle  $\mathbb{H}$  over  $X$.   Classical  results of  Seeley  [Se]  and  Atiyah  and  Singer  [$AS_{\hskip 0.35002pt4}$]  show  that  the  latter  family  is  fully  Fredholm.   Hence  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  applies  to such  families.

Suppose now  that  we are given a family of  elliptic self-adjoint  differential  operators of  order $1$ acting on  fibers of  $\mathbb{M}$.   This  is  the class of  families considered  in  [$MP_{\hskip 0.35002pt1}$],   Proposition  1.   Such a family defines a family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  of  closed densely defined operators  acting in  fibers of  a  Hilbert  bundle  $\mathbb{H}$  over  $X$.   It  is  well  known  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  has  the properties defining  discrete-spectrum  fully  Fredholm  families.   Hence  Corollary  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  applies  to such  families.

5. The second  proof

Compactly-polarized  operators.   Let  us  call  a self-adjoint  operator  $A\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  in a  Hilbert  space  $K$  compactly-polarized   if  $\|\hskip 1.99997ptA\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt1$  and  the essential  spectrum of  $A$  consists of  two points  $-\hskip 1.99997pt1\hskip 0.50003pt,\hskip 1.99997pt1$,   i.e.  $A$  belongs  to  Atiyah–Singer  [AS]  space  $\hat{F}_{*}$.   Such operator  $A$  is  automatically  Fredholm.   One can also define compactly-polarized operators as essentially unitary operators with  the norm $1$.   The  term  compactly-polarized   is  intended  to stress  the obvious analogy  with  finitely-polarized  operators from  Section  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza.

We claim  that  every compactly-polarized operator $A$  has  the form  $Q\hskip 1.99997pt+\hskip 1.99997ptk$,   where $Q$  is  a unitary self-adjoint  operator and $k$  is  a compact  self-adjoint  operator.   Of  course,   this  is  well  known,   but  we need some notations from  the proof.   Let  $A$  be a compactly-polarized operator and  let  $\varepsilon\hskip 1.99997pt\in\hskip 1.99997pt(\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt)$  be such  that  $(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced operator.   Let

Clearly,  $Q\hskip 3.99994pt=\hskip 3.99994ptf\hskip 1.49994pt(\hskip 1.49994ptA\hskip 1.49994pt)$  for some continuous function  $f\hskip 1.00006pt\colon\hskip 1.00006pt\mathbf{R}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{R}$  equal  to $1$ in a neighborhood of  $1$  and  to  $-\hskip 1.99997pt1$ in a neighborhood of  $-\hskip 1.99997pt1$.   Let  $q\hskip 0.50003pt,\hskip 1.99997pta$  be  the images of  $Q\hskip 0.50003pt,\hskip 1.99997ptA$  respectively  in  the  Calkin  algebra of  $H$.   Then  $q\hskip 3.99994pt=\hskip 3.99994ptf\hskip 1.49994pt(\hskip 1.49994pta\hskip 1.49994pt)$.   At  the same  time  the spectrum of  $a$  is  equal  to  the essential  spectrum of  $A$  and  hence consists of  two points  $-\hskip 1.99997pt1\hskip 0.50003pt,\hskip 1.99997pt1$.   It  follows  that  $a\hskip 3.99994pt=\hskip 3.99994ptf\hskip 1.49994pt(\hskip 1.49994pta\hskip 1.49994pt)$  and  hence  $a\hskip 3.99994pt=\hskip 3.99994ptq$.   In  turn,   this implies  that  $k\hskip 3.99994pt=\hskip 3.99994ptA\hskip 1.99997pt-\hskip 1.99997ptQ$  is  compact.

Compactly-polarized  families.   Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004ptx}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 3.00003ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  be a  family  of  self-adjoint  operators.   We will  say  that  such a family  is  a  compactly-polarized  family   if  it  is  fully  Fredholm  and  all  operators  $A_{\hskip 0.70004ptx}$  are compactly-polarized.

5.1. Lemma.   Suppose  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a compactly-polarized  family.   Then every  strictly adapted  ( local )  trivialization  is  fully  adapted.   

Proof.   Let  $z\hskip 1.99997pt\in\hskip 1.99997ptX$,  $\varepsilon\hskip 1.99997pt\in\hskip 1.99997pt(\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt)$,   and  $U_{\hskip 0.35002ptz}$  be a neighborhood of  $z$  such  that  $(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 0.50003pt,\hskip 1.99997pt\varepsilon\hskip 1.49994pt)$  is  an enhanced operator for every  $y\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$.   Suppose  that  we are given a strictly adapted  local  trivialization over  $U_{\hskip 0.35002ptz}$.   Such a  local  trivialization  turns  $P_{\hskip 1.39998pt[\hskip 0.70004pt\varepsilon\hskip 0.35002pt,\hskip 1.39998pt\infty\hskip 0.70004pt)}\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 0.50003pt,\hskip 1.99997pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family,   and  hence also  turns  $Q_{\hskip 1.39998pt\varepsilon}\hskip 1.00006pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 0.50003pt,\hskip 1.99997pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family.   It  is  fully adapted  if  and  only  if  it  turns  $A_{\hskip 0.70004pty}\hskip 0.50003pt,\hskip 1.99997pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family.   Since  $A_{\hskip 0.70004pty}\hskip 3.99994pt=\hskip 3.99994ptQ_{\hskip 1.39998pt\varepsilon}\hskip 1.00006pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 1.99997pt+\hskip 1.99997ptk_{\hskip 1.39998pt\varepsilon}\hskip 1.00006pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)$,   for strictly adapted  trivializations  the  latter condition  is  equivalent  to  turning  $k_{\hskip 1.39998pt\varepsilon}\hskip 1.00006pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 0.50003pt,\hskip 1.99997pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family.

Let  $K\hskip 1.49994pt(\hskip 1.49994ptH\hskip 1.49994pt)$  be  the space of  compact  operators  $H\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH$  with  the norm  topology.   By a  theorem of  Atiyah  and  Segal  [ASe]  the action of  the group  $\mathcal{U}\hskip 1.00006pt(\hskip 1.49994ptH\hskip 1.49994pt)$  on  $K\hskip 1.49994pt(\hskip 1.49994ptH\hskip 1.49994pt)$  by conjugations  is  continuous.   It  follows  that  if  the family  $k_{\hskip 1.04996pt\varepsilon}\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.99994pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  is  norm-continuous  in one  trivialization,   then  it  is  norm-continuous  in every  trivialization.

Since  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm,   the first  paragraph of  the proof  implies  that  for some  $U_{\hskip 0.35002ptz}$  some strictly adapted  trivialization over  $U_{\hskip 0.35002ptz}$  turns  $k_{\hskip 1.04996pt\varepsilon}\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.99994pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family.   Then by  the previous paragraph  every  trivialization over  $U_{\hskip 0.35002ptz}$  turns  $k_{\hskip 1.04996pt\varepsilon}\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004pty}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.99994pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family,   and  hence every strictly  adapted  trivialization  turns  the family  $A_{\hskip 0.70004pty}\hskip 0.50003pt,\hskip 1.99997pty\hskip 1.99997pt\in\hskip 1.99997ptU_{\hskip 0.35002ptz}$  into a norm continuous family.   The  lemma  follows.    $\blacksquare$

5.2. Theorem.   Suppose  that  the family  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  compactly-polarized.   Then a weak spectral  section  for  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  exists  if  and  only  if  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  vanishes.   

Proof.   As in  the proof  of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  we will  assume  that  $X$  is  compact.   By  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  there exists a strictly adapted  trivialization of  $\mathbb{H}$.   By  Lemma  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  such  trivialization  is  fully  adapted.   Hence  it  is  sufficient  to consider  norm-continuous  families  of  operators  in  a fixed  Hilbert  space  $H$.   In  this case  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  implies  that  a weak spectral  section exists  if  and  only  if  the family  is  homotopic  to a family of  invertible operators.   But  the  latter condition  is  equivalent  to  the vanishing of  the analytic index  (cf.  the proof  of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza).    $\blacksquare$

5.3. Theorem   Let  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  be a discrete-spectrum  and  fully  Fredholm  family.   Then a weak spectral  section  for  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  exists  if  and  only  if  its  analytic  index vanishes.   

Proof.   Since  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  fully  Fredholm,   the family  $C_{\hskip 0.70004ptx}\hskip 3.99994pt=\hskip 3.99994pt\gamma\hskip 1.49994pt(\hskip 1.49994ptA_{\hskip 0.70004ptx}\hskip 1.49994pt)\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  also fully  Fredholm.   Clearly,   every operator  $C_{\hskip 0.70004ptx}$  is  compactly-polarized.   It  follows  that  the family  $C_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  compactly-polarized.   By  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  the family  $C_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  admits a weak  spectral  section  if  and  only  if  its analytic index,   which  is  equal  to  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$,   vanishes.   At  the same  time every weak  spectral  section of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a weak  spectral  section of  $C_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$,   and  the converse  is  also  true.   The  theorem  follows.    $\blacksquare$

5.4. Corollary.   Suppose  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a discrete-spectrum  and  fully  Fredholm  family.   A spectral  section  for  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  exists  if  and  only  if  the analytic  index of  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  vanishes.

Proof.   It  is  sufficient  to combine  Theorems  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza.    $\blacksquare$

Remark.   While  Corollaries  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  are identical,   the proof  of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza,   in contrast  with  the proof  of  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza,   works only  for discrete-spectrum  families,   because  it  depends on  the properties of  compactly-polarized  families.

Remark.   As  the reader certainly  noticed,   we did  not  really  work with unbounded operators.   The definition of  spectral  sections  looks nicer  for unbounded operators,   but  can  be easily  reformulated  in  terms of  their bounded  transforms.   Strictly speaking,   Corollaries  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  are concerned  with  the image of  the bounded  transform,   i.e.  with  the families of  self-adjoint  strictly  contracting  operators.   But  in applications such families usually arise from  closed and densely defined  unbounded operators.

Remark.   Recent  results of  Prokhorova  [$P_{\hskip 0.35002pt3}$]  allow  to strengthen  the results of  this section.   Namely,   in  Lemma  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  it  is  sufficient  to assume  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a family of  compactly-polarized operators which  is  strictly  Fredholm  and such  that  $A_{\hskip 0.70004ptx}\hskip 1.99997pt-\hskip 1.99997pt\lambda\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a  Fredholm  family  for every  $\lambda\hskip 1.99997pt\in\hskip 1.99997pt(\hskip 1.00006pt-\hskip 1.99997pt1\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt)$.   In fact,   such families are automatically  fully  Fredholm.   See  [$P_{\hskip 0.35002pt3}$],   Theorem  5.   In  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza and  Corollary  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  it  is  sufficient  to assume  that  $A_{\hskip 0.70004ptx}\hskip 1.00006pt,\hskip 1.99997ptx\hskip 1.99997pt\in\hskip 1.99997ptX$  is  a discrete-spectrum  and  strictly  Fredholm  family.   As we pointed out  at  the end of  Section  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza,   the same  is  true for  Theorem  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza  and  Corollary  Spectral  sections :   two  proofs  of  a  theorem  of  Melrose–Piazza.   It  seems  that  strictly  Fredholm  families provide a proper context  for dealing with spectral  sections.