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Properties of m-complex symmetric operators

Muneo Chō, Eungil Ko, Ji Eun Lee

2017
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Studia Universitatis Babes-Bolyai Matematica
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In this paper, we study several properties of m-complex symmetric operators. In particular, we prove that if T ∈ L(H) is an m-complex symmetric operator and N is a nilpotent operator of order n > 2 with T N = N T , then T +N is a (2n+m−2)-complex symmetric operator. Moreover, we investigate the decomposability of T + A and T A where T is an m-complex symmetric operator and A is an algebraic operator. Finally, we provide various spectral relations of such operators. As some applications of these
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... plications of these results, we discuss Weyl type theorems for such operators. (2010) : 47A11, 47B25. Mathematics Subject Classification Keywords: Conjugation, m-complex symmetric operator, nilpotent perturbations, decomposable, Weyl type theorems. ( 1.2) Hence, if T is an m-complex symmetric operator with conjugation C, then T is an ncomplex symmetric operator with conjugation C for all n ≥ m. In sequel, it was shown from [10] that if m is even, then ∆ m (T ) is complex symmetric with the conjugation C, and if m is odd, then ∆ m (T ) is skew complex symmetric with the conjugation C. Moreover, we investigate conditions for (m + 1)-complex symmetric operators to be m-complex symmetric operators and characterize the spectrum of ∆ m (T ). All normal operators, algebraic operators of order 2, Hankel matrices, finite Toeplitz matrices, all truncated Toeplitz operators, some Volterra integration operators, nilpotent operators of order k, and nilpotent perturbations of Hermitian operators are included in the class of m-complex symmetric operators (see [14], [15], [16], [19], and [9] for more details). The class of m-complex symmetric operators is surprisingly large class. Many authors have studied Hermitian, isometric, unitary, and normal operators perturbed by nilpotent operators (see [2], [6], [8], and [21], etc). In 2014, T. Bermudez, A. Martinon, V. Muller, and J. Noda ([6]) have been studied the perturbation of misometries by nilpotent operators. In light of m-complex symmetric operators, we consider the nilpotent perturbations of m-complex symmetric operators. In particular, we prove that if T ∈ L(H) is an m-complex symmetric operator and N is a nilpotent operator of order n > 2 with T N = N T , then T + N is a (2n + m − 2)complex symmetric operator. Moreover, we investigate the decomposability of T + A

doi:10.24193/subbmath.2017.2.09
fatcat:wsyj4wlpijgjfeo346ynrmqfuq