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Register a new userGeneralized Browders theorem for tensor product and elementary operators
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The transfer property for the generalized Browders theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the $B$-Weyl spectrum inclusion. In addition, the isolated points of these two classes of operators will be fully characterized.
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The main objective of this work is to study generalized Browders and Weyls theorems for the multiplication operators $L_A$ and $R_B$ and for the elementary operator $tau_{A,B}=L_AR_B$.
A Banach space operator $Tin B(X)$ is left polaroid if for each $lambdainhbox{iso}sigma_a(T)$ there is an integer $d(lambda)$ such that asc $(T-lambda)=d(lambda)<infty$ and $(T-lambda)^{d(lambda)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-lambda)<infty$, $(T-lambda)X$ is closed and $dim(T-lambda)^{-1}(0)<infty$ at each $lambdainhbox{iso }sigma_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $tau_{AB}$, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$ if and only if $0 otinhbox{iso}sigma_a(Aotimes B)$; a similar result holds for $tau_{AB}$ for finitely left polaroid $A$ and $B^*$.
Given Banach spaces $X$ and $Y$ and operators $Ain B(X)$ and $Bin B(Y)$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $Aotimes Bin B(Xoverline{otimes} Y)$ or to the elementary operator defined by $A$ and $B$, $tau_{AB}=L_AR_Bin B(B(Y,X))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $Aotimes B$ and to $tau_{AB}$ will be given.
We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces and then provide an unified extension of the recent inequalities due to Choi [6], Lin [14] and Zhang et al. [5,19]. We demonstrate the applications of a positive semidefinite $3times 3$ block matrix, which motivates us to give a simple alternative proof of Dragomirs inequality and Kreins inequality.
Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martinez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural extreme cases that highlight the importance of which generalized Bregman distance is chosen.
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