Document Type
Article
Department
Mathematics (Pomona)
Publication Date
2007
Keywords
Complex symmetric operator, Takagi factorization, inner function, Aleksandrov-Clark operator, Clark operator, Aleksandrov measure, compressed shift, Jordan operator, $J$-selfadjoint operator, Sturm-Liouville problem
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-self adjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.
Rights Information
© 2007 American Mathematical Society
Terms of Use & License Information
Recommended Citation
Garcia, S.R., Putinar, M., Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913–3931. MR2302518 (2008b:47005)
Comments
First published in Transactions of the American Mathematical Society in Volume 359, Number 8, August 2007, published by the American Mathematical Society.