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Register a new userGeneralized Browders and Weyls theorems for left and right multiplication operators
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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 $tau_{A,B}=L_AR_B$.
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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.
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.
We consider a class of generalized nonexpansive mappings introduced by Karapinar [5] and seen as a generalization of Suzuki (C)-condition. We prove some weak and strong convergence theorems for approximating fixed points of such mappings under suitable conditions in uniformly convex Banach spaces. Our results generalize those of Khan and Suzuki [4] to the case of this kind of mappings and, in turn, are related to a famous convergence theorem of Reich [2] on nonexpansive mappings.
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.
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 $B^*$.
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