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- W1585548596 abstract "In this note we prove that if begin{equation*}M_{C}=left (begin {smallmatrix}A&C 0&Bend{smallmatrix} right ) end{equation*} is a $2times 2$ upper triangular operator matrix acting on the Banach space $Xoplus Y$, then $M_{C}$ is invertible for some $Cin mathcal {L}(Y,X)$ if and only if $Ain mathcal {L}(X)$ and $Bin mathcal {L}(Y)$ satisfy the following conditions: [(i)] $A$ is left invertible; [(ii)] $B$ is right invertible; [(iii)] $X/A(X)cong B^{-1}(0)$. Furthermore we show that $sigma (A)cup sigma (B)=sigma (M_{C})cup W$, where $W$ is the union of certain of the holes in $sigma (M_{C})$ which happen to be subsets of $sigma (A)cap sigma (B)$." @default.
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- W1585548596 date "1999-07-06" @default.
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- W1585548596 title "Invertible completions of $2times 2$ upper triangular operator matrices" @default.
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- W1585548596 doi "https://doi.org/10.1090/s0002-9939-99-04965-5" @default.
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