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It was recently shown in [24] that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we derive new cases where EAE and SC coincide and provide a new example for which EAE and SC do not coincide and where the Banach space are not essentially incomparable., Comment: 20 pages

Positive real odd matrix functions, often referred to as positive real lossless matrix functions, play an important role in many applications in multi-port electrical systems. In this paper we present closer analogues to some of the known results for the scalar, one-port, case in the multi-port setting. Specifically, we determine necessary and sufficient conditions for the well studied partial fraction formula to represent functions in the class of positive real odd matrix functions, and explicit minimal state space realization formulas for the inverse (admittance) of a function in this class, which itself is also a positive real odd matrix function. Doing so, enables us to provide a partial analogue of the poles-zero interlacing behavior from the scalar case., Comment: 21 pages

In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction and the case where the functions are matrix polynomials., Comment: 15 pages

This paper contains a further analysis of the Toeplitz-like operators $T_\omega$ on $H^p$ with rational symbol $\omega$ having poles on the unit circle that were previously studied in [5.6]. Here the adjoint operator $T_\omega^*$ is described. In the case where $p=2$ and $\omega$ has poles only on the unit circle $\mathbb{T}$, a description is given for when $T_\omega^*$ is symmetric and when $T_\omega^*$ admits a selfadjoint extension. Also in the case where $p=2$, $\omega$ has only poles on $\mathbb{T}$ and in addition $\omega$ is proper, it is shown that $T_\omega^*$ coincides with the unbounded Toeplitz operator defined by Sarason in [10]., Comment: 19 pages

For the strictly positive case (the suboptimal case) the maximum entropy solution $X$ to the Leech problem $G(z)X(z)=K(z)$ and $\|X\|_\infty=\sup_{|z|\leq 1}\|X(z)\|\leq 1$, with $G$ and $K$ stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for $X$ is given, and $\|X\|_\infty$ turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions $G$ and $K$. A formula for the entropy of $X$ is also given., Comment: 19 pages

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In the present paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions for compact Hilbert space operators to be equivalent after extension, in terms of their singular values, we prove, under certain additional conditions, the necessity of a similar relationship between the s-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary s-functions. We investigate equivalence after extension for operators on ℓp-spaces. We show that two operators that act on different ℓp-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension; however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces X and Y, we investigate consequences of an operator on X being equivalent after extension to a compact operator on Y. We show that, in this case, a closed finite codimensional subspace of Y must embed into X, and that certain general Banach space properties must transfer from X to Y. We also show that no operator on X can be equivalent after extension to an operator on Y, if X and Y are essentially incomparable Banach spaces.

A Redheffer type description of the set of all contractive solutions to the relaxed commutant lifting problem is given. The description involves a set of Schur class functions which is obtained by combining the method of isometric coupling with results on isometric realizations. For a number of special cases, including the case of the classical commutant lifting theorem, the description yields a proper parameterization of the set of all contractive solutions, but examples show that, in general, the Schur class function determining the contractive lifting does not have to be unique. Also some sufficient conditions are given guaranteeing that the corresponding relaxed commutant lifting problem has only one solution., Comment: Integral Equations Operator Theory 54 (2006), 33--67