Gershgorin’s theorem for matrices of operators. (English) Zbl 1018.47004
Summary: Let \(A= (A_{ij})\) be an \(n\times n\) matrix of operators acting on the Banach space \(X= X_1\oplus X_2\oplus\cdots\oplus X_n\) endowed with the \(\|\cdot\|_\infty\) norm. The Gershgorin circle theorem extends to this setting as follows: If
\[
G_i= \sigma(A_{ii})\cup \Biggl\{\lambda: \lambda\not\in \sigma(A_{ii})\text{ and }\|(\lambda- A_{ii})^{-1}\|^{-1}\leq \sum^n_{j=1, i\neq j}\|A_{ij}\|\Biggr\},
\]
then
\[
\sigma(A)\subset \bigcup^n_{i=1} G_i.
\]
Moreover, assume that \(J\) is a proper nonempty subset of \(\{1,2,\dots,n\}\), if \(\bigcup_{i\in J} G_i\) and \(\bigcup_{i\notin J} G_i\) are disjoint, then there exist invariant subspaces \(Y_1\) and \(Y_2\) for \(A\) such that
\[
\sigma(A|_{Y_1})\subset \bigcup_{i\in J} G_i\quad\text{and}\quad \sigma(A|_{Y_2})\subset \bigcup_{i\notin J} G_i,
\]
where \(Y_1\simeq \bigoplus_{i\in J} X_i\) and \(Y_2\simeq \bigoplus_{i\ni J}X_i\).
The notion of minimal Gershgorin sets that follows is one possible generalization of what Varga studied in the scalar case. (For partitioned matrices he used a more refined generalization.) For any positive \(n\)-dimensional vector \({\mathbf x}= (x_1,\dots, x_n)\), the operator \[ ((1/x_1)I\oplus\cdots\oplus (1/x_n)I) A(x_1 I\oplus\cdots\oplus x_n I)= A_{{\mathbf x}} \] is similar to \(A\). Let \[ G_{i,{\mathbf x}}(A)= \sigma(A_{ii})\cup \Biggl\{z: z\not\in \sigma(A_{ii})\text{ and }\|(A_{ii}- z)^{-1}\|^{-1}\leq \sum^n_{j=1,\;j\neq i}(x_j/x_i)\|A_{ij}\|\Biggr\}. \] The minimal Gershgorin set \(G(A)\) is defined to be \[ G(A)= \bigcap_{{\mathbf x}> 0} \bigcup^n_{i=1} G_{i,{\mathbf x}}(A). \] As in the scalar case, it has the property that \[ \sigma(\Omega_A)= \bigcup_{B\in\Omega_A} \sigma(B)\subset G(A), \] where \[ \Omega_A= \{B= (B_{ii}): B_{ii}= A_{ii}\text{ for }i= 1,\dots, n\text{ and }\|B_{ij}\|=\|A_{ij}\|\text{ if }i\neq j\}. \] It is proved that if each \(X_i\) is a Hilbert space and each \(A_{ii}\) is normal, \(i= 1,\dots, n\), then \[ \partial G(A)\subset \overline{\sigma(\Omega_A)}, \] where \(\partial G(A)\) denotes the boundary of \(G(A)\). It is worth remarking that the closure of \(\sigma(\Omega_A)\) is necessary.
The notion of minimal Gershgorin sets that follows is one possible generalization of what Varga studied in the scalar case. (For partitioned matrices he used a more refined generalization.) For any positive \(n\)-dimensional vector \({\mathbf x}= (x_1,\dots, x_n)\), the operator \[ ((1/x_1)I\oplus\cdots\oplus (1/x_n)I) A(x_1 I\oplus\cdots\oplus x_n I)= A_{{\mathbf x}} \] is similar to \(A\). Let \[ G_{i,{\mathbf x}}(A)= \sigma(A_{ii})\cup \Biggl\{z: z\not\in \sigma(A_{ii})\text{ and }\|(A_{ii}- z)^{-1}\|^{-1}\leq \sum^n_{j=1,\;j\neq i}(x_j/x_i)\|A_{ij}\|\Biggr\}. \] The minimal Gershgorin set \(G(A)\) is defined to be \[ G(A)= \bigcap_{{\mathbf x}> 0} \bigcup^n_{i=1} G_{i,{\mathbf x}}(A). \] As in the scalar case, it has the property that \[ \sigma(\Omega_A)= \bigcup_{B\in\Omega_A} \sigma(B)\subset G(A), \] where \[ \Omega_A= \{B= (B_{ii}): B_{ii}= A_{ii}\text{ for }i= 1,\dots, n\text{ and }\|B_{ij}\|=\|A_{ij}\|\text{ if }i\neq j\}. \] It is proved that if each \(X_i\) is a Hilbert space and each \(A_{ii}\) is normal, \(i= 1,\dots, n\), then \[ \partial G(A)\subset \overline{\sigma(\Omega_A)}, \] where \(\partial G(A)\) denotes the boundary of \(G(A)\). It is worth remarking that the closure of \(\sigma(\Omega_A)\) is necessary.
Keywords:
partitioned matrices; Riesz projections; normal operators; Hilbert and Banach spaces; Gershgorin circle theorem; invariant subspaces; minimal Gershgorin setsReferences:
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