Complex symmetric triangular operators

Complex Symmetric Operators and Applications

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, selfadjoint extensions of s...

Complex Symmetric Operators and Applications Ii

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 (compr...

Review: A C*-algebra approach to complex symmetric operators

Mathematical and physical aspects of complex symmetric operators

Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugatelinear symmetric operators...

On the Closure of the Complex Symmetric Operators: Compact Operators and Weighted Shifts

We study the closure CSO of the set CSO of all complex symmetric operators on a separable, infinite-dimensional, complex Hilbert space. Among other things, we prove that every compact operator in CSO is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in CSO\CSO. In particular, we show that weighted shifts which demonstrate a...