Structured Invariant Spaces of Vector Valued Functions, Hermitian Forms and a Generalization of Iohvidov\u27s Laws

Vector spaces of pairs of rational vector valued functions, which are (1) invariant under the generalized backward shift and (2) endowed with a sesquilinear form which is subject to a structural identity, are studied. It is shown that any matrix can be viewed as the “Gram” matrix of a suitably defined basis for such a space. This identification is used to show that a rule due to Iohvidov for evaluating the rank of certain subblocks of a Toeplitz (or Hankel) matrix is applicable to a wider class of matrices with (appropriately defined) displacement rank equal to two. Enroute, a theory of reproducing kernel spaces is developed for nondegenerate spaces of the type mentioned above

Norms of basic operators in vector valued model spaces and de Branges spaces

Let $\Omega_+$ be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator $A_\alpha=\Pi_\Theta T_{b_\alpha}|_{\mathcal{H}(\Theta)}$ in the vector valued model space $\mathcal{H}(\Theta)=H^m_2 \ominus \Theta H^m_2$ associated with an $m\times m$ matrix valued inner function $\Theta$ in $\Omega_+$ and show that the norm is attained. Here $\Pi_\Theta$ denotes the orthogonal projection from the Lebesgue space $L^m_2$ onto $\mathcal{H}(\Theta)$ and $T_{b_\alpha}$ is the operator of multiplication by the elementary Blaschke factor $b_{\alpha}$ of degree one with a zero at a point $\alpha\in \Omega_+$. We show that if $A_\alpha$ is strictly contractive, then its norm may be expressed in terms of the singular values of $\Theta(\alpha)$. We then extend this evaluation to the more general setting of vector valued de Branges spaces.Comment: 16 pages, Revised, Section 5 is new, to appear in Integral Equations and Operator Theor

Structured Invariant Spaces of Vector Valued Functions, Sesquilinear Forms and a Generalization of Iohvidov\u27s Laws

Finite dimensional indefinite inner product spaces of vector valued rational functions which are (1) invariant under the generalized backward shift and (2) subject to a structural identity, and subspaces and “superspaces” thereof are studied. The theory of these spaces is then applied to deduce a generalization of a pair of rules due to lohvidov for evaluating the inertia of certain subblocks of Hermitian Toeplitz and Hermitian Hankel matrices. The connecting link rests on the identification of a Hermitian matrix as the Gram matrix of a space of vector valued functions of the type considered in the first part of the paper. Corresponding generalizations of another pair of theorems by lohvidov on the rank of certain subblocks of non-Hermitian Teoplitz and non-Hermitian Hankel matrices are also stated, but the proofs will be presented elsewhere

On linear fractional transformations associated with generalized J-inner matrix functions

In this paper we study generalized J-inner matrix valued functions which appear as resolvent matrices in various indefinite interpolation problems. Reproducing kernel indefinite inner product spaces associated with a generalized J-inner matrix valued function W are studied and intensively used in the description of the range of the linear fractional transformation associated with W and applied to the Schur class. For a subclass of generalized J-inner matrix valued function W the notion of associated pair is introduced and factorization formulas for W are found.Comment: 41 page

On a New Class of Realization Formulas and their Application

A new set of realization formulas is derived for a class of matrix-valued functions W(λ). These include the standard realization formulas for rational W(λ). Observability, controllability, and minimality are defined and characterized. Conditions for W(λ) to be (J1, J2) isometric and/or (J1, J2) coisometric with respect to a pair of signature matrices J1 and J2 are given in terms of the realizations. Minimal factorizations for square W(λ) are considered, and formulas for the factors are deduced. Associated reproducing kernel spaces (of pairs too) are discussed

Criteria for the strong regularity of J-inner functions and γ-generating matrices

AbstractThe class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A2) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of γ-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. A determinental criterion is also presented