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A Banach space operator $Tin B({cal X})$ is polaroid if points $lambdainisosigmasigma(T)$ are poles of the resolvent of $T$. Let $sigma_a(T)$, $sigma_w(T)$, $sigma_{aw}(T)$, $sigma_{SF_+}(T)$ and $sigma_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $Cin B({cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $pi_0(M_C)={lambdainisosigma(M_C) 0<dim(M_C-lambda)^{-1}(0)<infty}$, $M_0$ satisfies Weyls theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$ and $B$ has SVEP at points $muinsigma_w(M_0)setminussigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$ and $B^*$ has SVEP at points $muinsigma_w(M_0)setminussigma_{SF_-}(B)$, then $sigma(M_C)setminussigma_w(M_C)=pi_0(M_C)$. Here the hypothesis that $lambdainpi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $lambdainpi_0(A)$ are poles of the resolvent of $A$. For an operator $Tin B(X)$, let $pi_0^a(T)={lambda:lambdainisosigma_a(T), 0<dim(T-lambda)^{-1}(0)<infty}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $pi_0^a(M)$ and $B$ is polaroid on $pi_0^a(B)$, then $sigma_a(M)setminussigma_{aw}(M)=pi_0^a(M)$.
Let $T=begin{bmatrix} X &Y 0 & Zend{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $det(I_n+T^*T)ge det(I_r+X^*X)cdot det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.
We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic bo
Given two complex Banach spaces $X_1$ and $X_2$, a tensor product of $X_1$ and $X_2$, $X_1tilde{otimes}X_2$, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, $S=(S_1,ldots ,S_n)$ and $T=(T_1,ldots ,T_m)$, defined on $
In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction of embedda
We obtain limit theorems for $Phi(A^p)^{1/p}$ and $(A^psigma B)^{1/p}$ as $ptoinfty$ for positive matrices $A,B$, where $Phi$ is a positive linear map between matrix algebras (in particular, $Phi(A)=KAK^*$) and $sigma$ is an operator mean (in particu