Some results on \((A; (m, n))\)-isosymmetric operators on a Hilbert space. (English) Zbl 1510.47031
Summary: In this paper, we introduce the class of \((A; (m, n))\)-isosymmetric operators and we study some of their properties, for a positive semi-definite operator \(A\) and \(m,n\in \mathbb{N}\), which extend, by changing the initial inner product with the semi-inner product induced by \(A\), the well-known class of \((m, n)\)-isosymmetric operators introduced by M. Stankus [Isosymmetric linear transformations on complex Hilbert space. San Diego: University of California (PhD Thesis) (1993); Integral Equations Oper. Theory 75, No. 3, 301–321 (2013; Zbl 1284.47015)]. In particular, we characterize a family of \(A\)-isosymmetric \((2\times 2)\) upper triangular operator matrices. Moreover, we show that, if \(T\) is \((A; (m, n))\)-isosymmetric and if \(Q\) is a nilpotent operator of order \(r\) doubly commuting with \(T\), then \(T^p\) is \((A; (m, n))\)-isosymmetric symmetric for any \(p\in \mathbb{N}\) and \((T +Q)\) is \((A;(m+2r -2, n+2r -1))\)-isosymmetric. Some properties of the spectrum are also investigated.
MSC:
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 47B65 | Positive linear operators and order-bounded operators |
| 47A55 | Perturbation theory of linear operators |
Keywords:
semi-Hilbert space; isosymmetric operators; \((A; (m, n))\)-isosymmetric operators; \((A, m)\)-isometric operators; \((A, m)\)-symmetric operators; spectrum; nilpotent perturbationCitations:
Zbl 1284.47015References:
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