Maps preserving the local spectral subspace of skew-product of operators

Throughout this paper, $H$ and $K$ are infinite-dimensional complex Hilbert spaces. As usual $B(H,K)$ denotes the space of all bounded linear operators from $H$ into $K$. When $H=K$ we simply write $B(H)$ instead of $B(H,H)$, and its unit will be denoted by $I$. The inner product of $H$ or $K$ will be denoted by $\left\langle,\right\rangle$ if there is no confusion. For an operator $T\in B(H,K)$, let $T^{\ast}$ denote as usual its adjoint. A preserver problem generally deals with characterizing those maps on some specific algebraic structures which preserve a particular subset, property or relation. This subject has a long history and its origins goes back well over a century to the so-called first linear preserver problem, due to Frobenius [11], that determines linear maps preserving the determinant of matrices. As we mentioned earlier, the main of this subject goal is to describe the general form of linear maps between two Banach algebras which preserve a certain property, or a certain class of elements, or a certain relation. One of the most famous related problems is Kaplansky’s problem [17] asking whether every surjective unital invertibility preserving linear map between two semisimple Banach algebras is a Jordan homomorphism. His question was motivated by two classical results, the result of Marcus and Moyls [18] on linear maps preserving eigenvalues of matrices and the Gleason-Kahane-Zelazko theorem [15, 16] stating that every unital invertibility preserving linear functional on a unital complex Banach algebra is necessarily multiplicative. The later this result was obtained independently by Gleason in [15] and Kahane-Zelazko in [16], and was refined by Zelazko in [23]. In the non-commutative case, the best known result so far are due to Sourour [22]. He answered to the Kaplansky’s question in the affirmative for bijective unital linear invertibility preserving maps acting on the algebra of all bounded operators on a Banach space. Note that when the maps are unital, then preserving invertibility is equivalent to preserving spectrum. These results opened the gate for many authors who investigate linear (or additive) maps preserving spectrum; see for instance [1, 12, 13] and the references therein. Along this line, Molnar [19] investigated maps preserving the spectrum of operator products without assuming linearity or additivity.

The local resolvent set, $\rho_{T}(x)$, of an operator $T\in B(H)$ at a point $x\in H$ is the union of all open subsets $U$ of the complex plane $\mathbb{C}$ for which there is an analytic function $f:U\longrightarrow H$ such that $(\mu I-T)f(\mu)=x$ for all $\mu\in U$. The complement of local resolvent set is called the local spectrum of $T$ at $x$, denoted by $\sigma_{T}(x)$, and is obviously a closed subset (possibly empty) of $\sigma(T)$, the spectrum of $T$. We recall that an operator $T\in B(H)$ is said to have the single-valued extension property (henceforth abbreviated to SVEP) if, for every open subset $U$ of $\mathbb{C}$, there exists no nonzero analytic solution, $f:U\longrightarrow H$, of the equation

Every operator $T\in B(H)$ for which the interior of its point spectrum, $\sigma_{p}(T)$, is empty enjoys this property.

For every subset $F\subseteq\mathbb{C}$ the local spectral subspace $H_{T}(F)$ is defined by

Clearly, if $F_{1}\subseteq F_{2}$ then $H_{T}(F_{1})\subseteq H_{T}(F_{2})$. For more information about these notions one may see the books [2, 20].

The study of linear and nonlinear local spectra preserver problems attracted the attention of a number of authors. Bourhim and Ransford were the first ones to consider this type of preserver problem, characterizing in [8] additive maps on $B(X)$, the algebra of all linear bounded operators on infinite-dimensional complex Banach space $X$, that preserves the local spectrum of operators at each vector of $X$. Their results motivated several authors to describe maps on matrices or operators that preserve local spectrum, local spectral radius, and local inner spectral radius; see, for instance, the last section of the survey article [5] and the references therein. Based on the results from the theory of linear preservers proved by Jafarian and Sourour [14], Dolinar et al. [9], characterised the form of maps preserving the lattice of sum of operators. They showed that the map (not necessarily linear) $\varphi:B(X)\rightarrow B(X)$ satisfies Lat$(\varphi(T)+\varphi(S))=$Lat$(T+S)$ for all $T,S\in B(X)$, if and only if there are a non zero scalar $\alpha$ and a map $\phi:B(X)\rightarrow\mathbb{F}$ such that $\varphi(T)=\alpha T+\phi(T)I$ for all $T\in B(X)$ (See [9, Theorem 1]), where $\mathbb{F}$ is the complex field $\mathbb{C}$ or the real field $\mathbb{R}$ and Lat$(T)$ is denoted the lattice of $T$, that is, the set of all invariant subspaces of $T$. They proved also, in the same paper, that a not necessarily linear maps $\varphi:B(X)\rightarrow B(X)$ satisfies Lat$(\varphi(T)\varphi(S))=$ Lat$(TS)$ (resp. Lat$(\varphi(T)\varphi(S)\varphi(T))=$ Lat$(TST)$, resp. Lat$(\varphi(T)\varphi(S)+\varphi(S)\varphi(T))=$ Lat$(TS+ST)$) for all $T,S\in B(X)$, if and only if there is a map $\phi:B(X)\rightarrow\mathbb{F}$ such that $\varphi(T)\neq 0$ if $T\neq 0$ and $\varphi(T)=\phi(T)T$ for all $T\in B(X)$ (See [9, Theorem 2]).

For a Banach space $X$, it is well-known that $X_{T}(F)$, the local spectral subspace of $T$ associated with a subset $F$ of $\mathbb{C}$, is an element of $Lat(T)$, so one can replace the lattice preserving property by the local spectral subspace preserving property. In [10], the authors described additive maps on $B(X)$ that preserve the local spectral subspace of operators associated with any singleton. More precisely, they proved that the only additive map $\varphi$ on $B(X)$ for which $X_{\varphi(T)}(\{\lambda\})=X_{T}(\{\lambda\})$ for all $T\in B(X)$ and $\lambda\in\mathbb{C}$, is the identity. In [4], Benbouziane et al. characterized the forms of surjective weakly continuous maps $\varphi$ from $B(X)$ into $B(X)$ which satisfy

Afterwards, in [3], the authors studied surjective maps that preserve the local spectral subspace of the sum of two operators associated with non-fixed singletons. In other words, they characterized surjective maps $\varphi$ on $B(X)$ which satisfy

They also gave a characterization of maps on $B(X)$ that preserve the local spectral subspace of the difference of operators associated with non-fixed singletons. Furthermore, they investigated the product case as well as the triple product case. Namely, they described surjective maps $\varphi$ on $B(X)$ satisfying

and also surjective maps $\varphi$ on $B(X)$ satisfying

Bourhim and Lee [6] investigated the form of all maps $\varphi_{1}$ and $\varphi_{2}$ on $B(X)$ such that, for every $T$ and $S$ in $B(X)$, the local spectra of $TS$ and $\varphi_{1}(T)\varphi_{2}(S)$ are the same at a nonzero fixed vector $x_{0}$. In this paper, We show that if $\varphi:B(H)\rightarrow B(H)$ is an additive map such that its range contains all operators of rank at most two and satisfies

then there exist a unitary operator $V$ in $B(H)$ and a nonzero scalar $\mu$ such that $\varphi(T)=\mu TV^{\ast}$ for all $T\in B(H)$. We also investigate the form of all maps $\varphi_{1}$ and $\varphi_{2}$ on $B(H)$ such that, for every $T$ and $S$ in $B(H)$, the local spectral subspaces of $TS^{\ast}$ and $\varphi_{1}(T)\varphi_{2}(S)^{\ast}$, associated with the singleton $\{\lambda\}$, coincide.

The first lemma summarizes some known basic and properties of the local spectrum.

In the next theorem we collect some of the basic properties of the subspaces $H_{T}(F)$.

For two nonzero vectors $x$ and $y$ in $H$, let $x\otimes y$ stands for the operator of rank at most one defined by

Note that every rank one operator in $B(H)$ can be written in this form, and that every finite rank operator $T\in B(H)$ can be written as a finite sum of rank one operators; i.e., $T=\sum_{i=1}^{n}x_{i}\otimes y_{i}$ for some $x_{i},y_{i}\in H$ and $i=1,2,...,n$. By $F(H)$ and $F_{n}(H)$, we mean the set of all finite rank operators in $B(H)$. and the set of all operators of rank at most $n$, $n$ is a positive integer, respectively.

The following lemma is an elementary observation which discribes the nonzero local spectrum of any rank one operator.

The following theorem, which may be of independent interest, gives a spectral characterization of rank one operators in term of local spectrum.

The following Lemma is a key tool for the proofs in the sequel.

Moreover, this theorem will be useful in the proofs of our main result.

The next theorem describes additive maps on $B(H)$ that preserve the local spectral subspace of operators associated with any singleton.

The following theorem will be useful in the sequel. We recall that if $h:\mathbb{C}\rightarrow\mathbb{C}$ is a ring homomorphism, then an additive map $A:H\rightarrow H$ satisfying $A(\alpha x)=h(\alpha)x,(x\in H,\alpha\in\mathbb{C})$ is called an $h$-quasilinear operator.

Note that, if in Theorem 2.8 the map $\varphi$ is linear, then $h$ is the identity map on $\mathbb{C}$ and so the maps $A,B,C$ and $D$ are linear.

The following theorem is the first main result of this paper, which characterizes those maps preserving the local spectral subspace of skew-product operators.

From this result, it is easy to deduce a generalization for the case of two different Hilbert spaces $H,K$.

In the next theorem, we investigate the form of all maps $\varphi_{1}$ and $\varphi_{2}$ on $B(H)$ such that, for every $T$ and $S$ in $B(H)$, the local spectral subspaces of $TS^{\ast}$ and $\varphi_{1}(T)\varphi_{2}(S)^{\ast}$, associated with the singleton $\{\lambda\}$, coincide.