Inverse-closedness of the subalgebra of locally nuclear operators. (English) Zbl 07722723
Summary: Let \(X\) be a Banach space and \(T\) be a bounded linear operator acting in \(l_p(\mathbb{Z}^c,X), 1 \le p\le\infty\). The operator \(T\) is called locally nuclear if it can be represented in the form \[(Tx)_k=\sum_{m\in\mathbb{Z}^c}b_{km}x_{k-m},\ k\in\mathbb{Z}^c,\] where \(b_{km}:X\to X\) are nuclear, \[\vert \vert b_{km}\vert \vert_{\mathfrak S_1}\le \beta_m,\ k,m\in\mathbb{Z}^c,\] \(\vert \vert \cdot\vert \vert_{\mathfrak S_1}\) is the nuclear norm, \(\beta\in l_g(\mathbb{Z}^c,X)\), and \(g\) is an appropriate weight on \(\mathbb{Z}^c\). It is established that, if \(T\) is locally nuclear and the operator \((1+T)^{-1}\) is invertible, then the inverse operator \((1 + T)^{-1}\) has the form \(1 + T_1\), where \(T_1\) is also locally nuclear. This result is refined for the case of operators acting in \(l_p(\mathbb{R}^c,\mathbb{C})\).
MSC:
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 35P05 | General topics in linear spectral theory for PDEs |
Keywords:
full subalgebra; inverse-closedness; nuclear operator; difference operator; convolution operator; weighted normReferences:
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