Some new classes of complex symmetric operators

Abstract

Abstract. We say that an operator T ∈ B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C : H → H so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T * ).

Properties of <i>m</i>-complex symmetric operators

“…Let R ∈ L(H ⊕ H) be a 2-normal operator. Then R is complex symmetric from [16] and R is unitarily equivalent to Let us recall that for an operator T ∈ L(H), a closed subspace M ⊂ H is invariant for T if T M ⊂ M, and it is hyperinvariant for T if it is invariant for every operator in the commutant {T } = {S ∈ L(H) : T S = ST } of T . A subspace M of H is nontrivial if it is different from {0} and H. As some applications of Theorem 3.9, we get the following corollary.…”

“…Set ∆ m (T ) := 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.…”

On ∞-Complex Symmetric Operators

“…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 operator are included in the class of m-complex symmetric operators. We refer the reader to [5][6][7][8]10, 11], and [2] for more details. The class of ∞-complex symmetric operators is the large class which contains finite-complex symmetric operators.…”

Aluthge Transforms of Complex Symmetric Operators

“…W. Wogen and the author have recently proved a generalization of Theorem 5. In particular, they show that every operator which is algebraic of degree two is complex symmetric [21].…”