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Upper triangular matrix operators with diagonal \((T_1,T_2)\), \(T_2\) \(k\)-nilpotent

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Abstract

If \(T=\left(\begin{array}{clcr}T_1&\quad C\\ 0&\quad T_2\end{array}\right) \in B(\mathcal{X }_1\oplus \mathcal{X }_2)\) is a Banach space upper triangular operator matrix with diagonal \((T_1, T_2)\) such that \(T_2\) is \(k\)-nilpotent for some integer \(k\ge 1\), then \(T\) inherits a number of its spectral properties, such as SVEP, Bishop’s property \((\beta )\) and the equality of Browder and Weyl spectrum, from those of \(T_1\). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as \(k\)-quasi hyponormal, \(k\)-quasi isometric and \(k\)-quasi paranormal operators, defined by a positivity condition.

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    B. P. Duggal

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Duggal, B.P. Upper triangular matrix operators with diagonal \((T_1,T_2)\), \(T_2\) \(k\)-nilpotent. Rend. Circ. Mat. Palermo 62, 215–226 (2013). https://doi.org/10.1007/s12215-013-0104-z

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