Abstract
In this paper, we introduce the class of [m]-complex symmetric operators and study various properties of this class. In particular, we show that if T is an [m]-complex symmetric operator, then T^{n} is also an [m]-complex symmetric operator for any ninmathbb {N}. In addition, we prove that if T is an [m]-complex symmetric operator, then sigma_{a}(T), sigma_{mathrm{SVEP}}(T), sigma_{beta }(T), and sigma_{(beta)_{epsilon}}(T) are symmetric about the real axis. Finally, we investigate the stability of an [m]-complex symmetric operator under perturbation by nilpotent operators commuting with T.
Highlights
Let H be a complex separable Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H
Several authors have studied the structure of a complex symmetric operator
If T is nilpotent of order k, T is a [2k – 1]-complex symmetric operator with conjugation C
Summary
Let H be a complex separable Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. An operator T ∈ B(H) is said to be complex symmetric if T∗ = CTC. Several authors have studied the structure of a complex symmetric operator.
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