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Register a new userTensor products and the semi-Browder joint spectra
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Authors
Enrico Boasso
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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 $X_1$ and $X_2$ respectively, we consider the $(n+m)$-tuple of operators defined on $X_1tilde{otimes}X_2$, $(Sotimes I,Iotimes T)= (S_1otimes I,ldots ,S_notimes I,Iotimes T_1,ldots ,I otimes T_m)$, and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. Muller and V. Rako$check{c}$evi$acute{ c}$ in [7] and of the split semi-Browder joint spectra (see section 3), of the $(n+m)$-tuple $(Sotimes I ,Iotimes T)$, in terms of the corresponding joint spectra of $S$ and $T$. This result is in some sense a generalization of a formula obtained for other various Browder spectra in Hilbert spaces and for tensor products of operators and for tuples of the form $(Sotimes I ,Iotimes T)$. In addition, we also describe all the mentioned joint spectra for a tuple of left and right multiplications defined on an operator ideal between Banach spaces in the sense of [5].
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Given two complex Banach spaces $X_1$ and $X_2$, a tensor product $X_1tilde{otimes} X_2$ of $X_1$ and $X_2$ in the sense of [14], two complex solvable finite dimensional Lie algebras $L_1$ and $L_2$, and two representations $rho_icolon L_ito {rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1times L_2$, and the tensor product representation of $L$, $rhocolon Lto {rm L}(X_1tilde{otimes}X_2)$, $rho=rho_1otimes I +Iotimes rho_2$. In this work we study the S{l}odkowski and the split joint spectra of the representation $rho$, and we describe them in terms of the corresponding joint spectra of $rho_1$ and $rho_2$. Moreover, we study the essential S{l}odkowski and the essential split joint spectra of the representation $rho$, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of $rho_1$ and $rho_2$. In addition, with similar arguments we describe all the above-mentioned joint spectra for the multiplication representation in an operator ideal between Banach spaces in the sense of [14]. Finally, we consider nilpotent systems of operators, in particular commutative, and we apply our descriptions to them.
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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 $sigma_a(M)setminussigma_{aw}(M)=pi_0^a(M)$.
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We show that Kraus property $S_{sigma}$ is preserved under taking weak* closed sums with masa-bimodules of finite width, and establish an intersection formula for weak* closed spans of tensor products, one of whose terms is a masa-bimodule of finite width. We initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form $kappa times lambda$, where $kappa$ is a set of finite width, while $lambda$ is operator synthetic, is, under a necessary restriction on the sets $lambda$, again operator synthetic. We show that property $S_{sigma}$ is preserved under spatial Morita subordinance. En route, we prove that non-atomic ternary masa-bimodules possess property $S_{sigma}$ hereditarily.
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