A class of operators related to $m$-symmetric operators


Abstract: $m$-symmetric operator plays a crucial role in the development of operator theory and has been widely studied due to unexpected intimate connections with differential equations, particularly conjugate point theory and disconjugacy. For positive integers $m$ and $k$, an operator $T$ is said to be a $k$-quasi-$m$-symmetric operator if $T^{*k}(\sum\limits_{j=0}^{m}(-1)^{j}(^{m}_{j})T^{*m-j}T^{j})T^{k}=0$, which is a generalization of $m$-symmetric operator. In this paper, some basic structural properties of $k$-quasi-$m$-symmetric operators are established with the help of operator matrix representation. In particular, we also show that every $k$-quasi-$3$-symmetric operator has a scalar extension. Finally, we prove that generalized Weyl's theorem holds for $k$-quasi-$3$-symmetric operators.

Keywords: $K$-quasi-$m$-symmetric operator, subscalarity, hyperinvariant subspace, Weyl's theorem

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