Variational principles for symmetric bilinear forms

Abstract

Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions.

Mathematical and physical aspects of complex symmetric operators

“…The following result demonstrates the nature of Friedrichs inequality at the abstract level [35]. Theorem 7.20.…”

An extremal problem for characteristic functions

“…Although much of the following can be phrased in terms of large truncated Toeplitz matrices [3], the arguments involve reproducing kernels and conjugations which are more natural in the setting of truncated Toeplitz operators [4,6,13]. For n ≥ 1, a simple computation with Fourier series shows that…”

Recent Progress on Truncated Toeplitz Operators

“…In particular, Theorem 12 says each truncated Toeplitz operator is a complex symmetric operator, a class of Hilbert space operators which has undergone much recent study [24,35,[45][46][47][48][49][50]54,55,61,63,71,72,74,[103][104][105][106][107]. In fact, it is suspected that truncated Toeplitz operators might serve as some sort of model operator for various classes of complex symmetric operators (see Section 9).…”

“…Consequently, one sees that M = gK u is a (closed) nearly invariant subspace of H 2 with extremal function g as in (35). The next natural step towards defining truncated Toeplitz operators on the nearly invariant subspace M = gK u is to understand P M , the orthogonal projection of L 2 onto M. The following lemma from [65] provides an explicit formula relating P M and P u .…”