Complex symmetric operators and applications II

Abstract

Abstract. A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT * C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ|T |, where J is an auxiliary conjugation commuting with |T | = √ T * T . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = CJ|T | also extends to the class of unbounded C-selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

Some new classes of complex symmetric operators

“…We refer the reader to [7,8] (or [9] for a more expository pace) for further details. Other recent articles concerning complex symmetric operators include [3,11].…”

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.…”

Variational principles for symmetric bilinear forms

“…It includes all normal operators, Hankel operators, compressed Toeplitz operators (including the compressed shift), and many standard integral operators such as the Volterra operator (see [9,10] for more examples). In the unbounded context, somewhat confusingly, C-symmetric operators are sometimes referred to as J-selfadjoint, although this should not be confused with the notion of J-selfadjointness arising in the theory of Krein spaces.…”

“…Instead of appealing to Takagi's theorem, we require a recent refinement of the polar decomposition for complex symmetric operators due to the second and third authors (see [10]). Recall that the polar decomposition T = U |T | of an operator T expresses T uniquely as the product of a positive operator |T | = √ T * T and a partial isometry U which satisfies ker U = ker |T | and maps cl ran |T | onto cl ran T .…”

“…In particular, the linear operator J 2 is the orthogonal projection onto the closed subspace ran J = (ker J) ⊥ . The following lemma, whose proof we briefly sketch, is from [10]: Lemma 2.7 If T : H → H is a bounded C-symmetric operator, then T = CJ|T | where J is a partial conjugation, supported on cl ran |T |, which commutes with |T | = √ T * T .…”