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By interpreting the well-known Brown–Halmos theorem for Toeplitz operators in terms of multipliers, we formulate a Brown–Halmos analogue for the product of generalized Toeplitz operators, defined as compressions of multiplication operators to closed subspaces of L2(T). We use this to define equivalences between two operators in that class by means of multipliers between the spaces where they act. Necessary and sufficient conditions for such an equivalence to be unitary or a similarity are established. The results are applied to Toeplitz and Hankel operators, truncated Toeplitz operators, and dual truncated Toeplitz operators.

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators.

A generalization to N×Nof the 2×2 Daniele–Khrapkov class of matrix‐valued functions is proposed. This class retains some of the features of the 2×2 Daniele–Khrapkov class, in particular, the presence of certain square‐root functions in its definition. Functions of this class appear in the study of finite‐dimensional integrable systems. The paper concentrates on giving the main properties of the class, using them to outline a method for the study of the Wiener–Hopf factorization of the symbols of this class. This is done through examples that are completely worked out. One of these examples corresponds to a particular case of the motion of a symmetric rigid body with a fixed point (Lagrange top). Copyright © 2001 John Wiley & Sons, Ltd.

A generalization to N×N of the 2×2 Daniele–Khrapkov class of matrix-valued functions is proposed. This class retains some of the features of the 2×2 Daniele–Khrapkov class, in particular, the presence of certain square-root functions in its definition. Functions of this class appear in the study of finite-dimensional integrable systems. The paper concentrates on giving the main properties of the class, using them to outline a method for the study of the Wiener–Hopf factorization of the symbols of this class. This is done through examples that are completely worked out. One of these examples corresponds to a particular case of the motion of a symmetric rigid body with a fixed point (Lagrange top). Copyright © 2001 John Wiley & Sons, Ltd.

The Wiener–Hopf factorization of a class of 2×2 symbols including matrices of Daniele–Khrapkov type is studied. The partial indices and the factors are determined, both in the canonical and non‐canonical cases. A non‐linear method is used which reduces the solution of a homogeneous Riemann–Hilbert problem to a non‐linear scalar equation. Copyright © 1999 John Wiley & Sons, Ltd.

The Wiener–Hopf factorization of a class of 2×2 symbols including matrices of Daniele–Khrapkov type is studied. The partial indices and the factors are determined, both in the canonical and non-canonical cases. A non-linear method is used which reduces the solution of a homogeneous Riemann–Hilbert problem to a non-linear scalar equation. Copyright © 1999 John Wiley & Sons, Ltd.