The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called k -quasi- m , n -isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the m , n -isosymmetric operators. We give a characterization for any operator to be k -quasi- m , n -isosymmetric operator. Using this characterization, we prove that any power of an k -quasi- m , n -isosymmetric operator is also an k -quasi- m , n -isosymmetric operator. Furthermore, we study the perturbation of an k -quasi- m , n -isosymmetric operator with a nilpotent operator. The product and tensor products of two k -quasi- m , n -isosymmetries are investigated.
\scrB (\scrH ) will denote the algebra of all bounded linear operators on a complex Hilbert space \scrH . In [6], the authors proved that natural power of a posinormal operator is not in general posinormal. Precisely, they constructed an example of a posinormal operator with square not being posinormal. Given a positive integer n, the aim of this article is to study a class of operators in \scrB (\scrH ) called n-power-posinormal. This class is invariant under natural power and contains any natural power of any posinormal operator and all n-power normal operators.Позначимо через \scrB (\scrH ) алгебру всiх обмежених лiнiйних операторiв у комплексному гiльбертовiм просторi \scrH . У [6] доведено, що цiлий степiнь позiнормального оператора не обов'язково є позiнормальним. Зокрема, був наведений приклад позiнормального оператора, квадрат якого не є позiнормальним. Метою цiєї статтi є дослiдження класу n-степенево позiнормальних операторiв з \scrB (\scrH ), iнварiантного вiдносно натуральних степенiв, який мiстить натуральнi степенi позiнормальних операторiв та n-степенево нормальнi оператори. 2020 Mathematics Subject Classification. 47B20.
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