10.4064/sm180714-12-3

We study analytic and geometric properties of Stieltjes and inverse Stieltjes families defined on a separable Hilbert space and establish various minimal representations for them by means of compressed resolvents of various types of linear relations. Also attention is paid to some new peculiar properties of Stieltjes and inverse Stieltjes families, including an analog for the notion of inner functions which will be characterized in an explicit manner. In addition, families which admit different types of scale invariance properties are described. Two transformers that naturally appear in the Stieltjes and inverse Stieltjes classes are introduced and their fixed points are identified.fi=vertaisarvioitu|en=peerReviewed

Holomorphic operator-valued functions generated by passive selfadjoint systems

Let M be a Hilbert space. In this paper we study a class RS(m) of operator functions that are holomorphic in the domain C∖{(−∞,−1] ∪ [1,+∞)} and whose values are bounded linear operators in m . The functions in RS(m) are Schur functions in the open unit disk D and, in addition, Nevanlinna functions in C+∪C− . Such functions can be realized as transfer functions of minimal passive selfadjoint discrete-time systems.We give various characterizations for the class RS(m) and obtain an explicit form for the inner functions from the class RS(m) as well as an inner dilation for any function from RS(m) . We also consider various transformations of the class RS(m) , construct realizations of their images, and find corresponding fixed points.fi=vertaisarvioitu|en=peerReviewed

The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem

We prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \epsilon I_{\mathcal{H}}$ for some $\epsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=\bar{-\Delta|_{C_0^\infty(\Omega)}}$ in $L^2(\Omega; d^n x)$ for $\Omega\subset\mathbb{R}^n$ an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian $S_K$ (i.e., the Krein--von Neumann extension of $S$), $$S_K v = \lambda v, \quad \lambda \neq 0,$$ is in one-to-one correspondence with the problem of {\em the buckling of a clamped plate}, $$(-\Delta)^2u=\lambda (-\Delta) u \text{in} \Omega, \quad \lambda \neq 0, \quad u\in H_0^2(\Omega),$$ where $u$ and $v$ are related via the pair of formulas $$u = S_F^{-1} (-\Delta) v, \quad v = \lambda^{-1}(-\Delta) u,$$ with $S_F$ the Friedrichs extension of $S$. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).Comment: 16 page

Holomorphic operator valued functions generated by passive selfadjoint systems

In this paper we study a class $\mathcal R\mathcal S(\mathfrak M)$ of operator functions that are holomorphic in the domain $\mathbb C\setminus\{(-\infty,-1]\cup [1,+\infty)\}$ and whose values are contractive operators in a Hilbert space $(\mathfrak M)$. The functions in $\mathcal R\mathcal S(\mathfrak M)$ are Schur functions in the open unit disk $\mathbb D$ and, in addition, Nevanlinna functions in $\mathbb C_+\cup\mathbb C_-$. Such functions can be realized as transfer functions of minimal passive selfadjoint discrete-time systems. We give various characterizations for the class $\mathcal R\mathcal S(\mathfrak M)$ and obtain an explicit form for the inner functions from the class $\mathcal R\mathcal S(\mathfrak M)$ as well as an inner dilation for any function from $\mathcal R\mathcal S(\mathfrak M)$. We also consider various transformations of the class $\mathcal R\mathcal S(\mathfrak M)$, construct realizations of their images, and find corresponding fixed points.Comment: Version 2 with 35 pages where Section 6.6 has been adde

Contractions with rank one defect operators and truncated CMV matrices

AbstractThe main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some Hilbert space of dimension N⩽∞. These models complement nicely the well-known models of Livšic and Sz.-Nagy–Foias. We show that each such operator acting on some finite-dimensional (respectively, separable infinite-dimensional Hilbert space) is unitarily equivalent to some finite (respectively semi-infinite) truncated CMV matrix obtained from the “full” CMV matrix by deleting the first row and the first column, and acting in CN (respectively ℓ2(N)). This result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to Cantero, Moral and Velázquez, as well as an analog for contraction operators of the result from [Yu. Arlinskiĭ, E. Tsekanovskiĭ, Non-self-adjoint Jacobi matrices with a rank-one imaginary part, J. Funct. Anal. 241 (2006) 383–438] concerning dissipative non-self-adjoint operators with a rank one imaginary part. It is shown that another functional model for contractions with rank one defect operators takes the form of the compression f(ζ)→PK(ζf(ζ)) on the Hilbert space L2(T,dμ) with a probability measure μ onto the subspace K=L2(T,dμ)⊖C. The relationship between characteristic functions of sub-matrices of the truncated CMV matrix with rank one defect operators and the corresponding Schur iterates is established. We develop direct and inverse spectral analysis for finite and semi-infinite truncated CMV matrices. In particular, we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving Schur parameters. It is pointed out that if the mixed spectral data contains zero eigenvalue, then no solution, unique solution or infinitely many solutions may occur in the inverse problem for truncated CMV matrices. The uniqueness theorem for recovered truncated CMV matrix from the given mixed spectral data is established. In this part the paper is closely related to the results of Hochstadt and Gesztesy–Simon obtained for finite self-adjoint Jacobi matrices

Passive systems with a normal main operator and quasi-selfadjoint systems

Passive systems $\tau={T,M,N,H}$ with $M$ and $N$ as an input and output space and $H$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\tau$ with $M=N$ is said to be quasi-selfadjoint if $ran(T-T^*)\subset N$. The subclass $S^{qs}$ of the Schur class $S$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass $S^{qs}$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass $S^{qs}$ and the $Q$-function of $T$ is given.Comment: 29 page