Yu. M. Arlinskii

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Articles: 8

Representations of closed quadratic forms associated with Stieltjes and inverse Stieltjes holomorphic families of linear relations

Yu. M. Arlinski, S. Hassi

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 27 (2021), no. 2, 103-129

In this paper holomorphic families of linear relations that belong to the Stieltjes or inverse Stieltjes class are studied. It is shown that in their domain of holomorphy $\mathbb{C}\setminus\mathbb{R}_+$ the values of Stieltjes and inverse Stieltjes families are, up to a rotation, maximal sectorial. This leads to a study of the associated closed sesquilinear forms and their representations. In particular, it is shown that the Stieltjes and inverse Stieltjes holomorphic families of linear relations are of type (B) in the sense of Kato. These results are proved by using linear fractional transforms which connect these families to holomorphic functions that belong to a combined Nevanlinna-Schur class and a key tool then relies on a specific structure of contractive operators.

Розглядаються голоморфні сім’ї лінійних відношень, які належать до класу Стілтьєса та оберненого класу Стілтьєса. Показано, що в їхній області голоморфності $\mathbb{C}\setminus\mathbb{R}_+$ значення цих сімей є, з точністю до обертання, максимальними секторіальними. Із цим пов’язане дослідження відповідних замкнених півторалінійних форм та їхніх представлень. Зокрема, показано, що стілтьєсівські та обернені стілтьєсівські голоморфні сім’ї лінійних відношень належать до типу (В) у сенсі Като. Доведення базується на використанні дробово-лінійних перетворень, які переводять розглядувані сім’ї в голоморфні функції класу Неванлінни-Шура, псля чого використовується спеціальні структури операторів стиску.

Transformations of Nevanlinna operator-functions and their fixed points

Yu. M. Arlinskiĭ

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 23 (2017), no. 3, 212-230

We give a new characterization of the class ${\bf N}^0_{\mathfrak M}[-1,1]$ of the operator-valued in the Hilbert space ${\mathfrak M}$ Nevanlinna functions that admit representations as compressed resolvents ($m$-functions) of selfadjoint contractions. We consider the auto\-morphism ${\bf \Gamma}:$ $M(\lambda){\mapsto}M_{{\bf \Gamma}} (\lambda):=\left((\lambda^2-1)M(\lambda)\right)^{-1}$ of the class ${\bf N}^0_{\mathfrak M}[-1,1]$ and construct a realization of $M_{{\bf \Gamma}}(\lambda)$ as a compressed resolvent. The unique fixed point of ${\bf\Gamma}$ is the $m$-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation ${\bf\widehat \Gamma}:$ ${\mathcal M}(\lambda)\mapsto {\mathcal M}_{{\bf\widehat \Gamma}}(\lambda) :=-({\mathcal M}(\lambda)+\lambda I_{\mathfrak M})^{-1}$ that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point ${\mathcal M}_0$ of ${\bf\widehat \Gamma}$ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator ${\bf\widehat J}_0$ in the Hilbert space ${\bf H}_0=\ell^2({\mathbb N}_0)\bigotimes{\mathfrak M}$. We prove that ${\mathcal M}_0$ is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations $\{{\mathcal M}_{n+1}(\lambda)=-({\mathcal M}_n(\lambda)+\lambda I_{\mathfrak M})^{-1}\}$ of ${\bf\widehat\Gamma}$. We show that the pair $\{{\bf H}_0,{\bf \widehat J}_0\}$ is the inductive limit of the sequence of realizations $\{\widehat{\mathfrak H}_n,\widehat A_n\} $ of $\{{\mathcal M}_n\}$. In the scalar case $({\mathfrak M}={\mathbb C})$, applying the algorithm of I.S. Kac, a realization of iterates $\{{\mathcal M}_n(\lambda)\}$ as $m$-functions of canonical (Hamiltonian) systems is constructed.

Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them

Yu. M. Arlinskiĭ, S. Hassi

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Methods Funct. Anal. Topology 21 (2015), no. 3, 199-224

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space $\mathfrak H$ with an exit in some larger Hilbert space $\mathfrak H\oplus\mathcal H$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Krein-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.

Factorizations of nonnegative symmetric operators

Yury Arlinskiĭ, Yury Kovalev

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Methods Funct. Anal. Topology 19 (2013), no. 3, 211-226

We prove that each closed denselydefined and nonnegative symmetric operator $\dot A$ having disjointnonnegative self-adjoint extensions admits infinitely manyfactorizations of the form $\dot A=\mathcal L\mathcal L_0$, where $\mathcal L_0$ is aclosed nonnegative symmetric operator and $\mathcal L$ its nonnegativeself-adjoint extension. The same factorizations are also establishedfor a non-densely defined nonnegative closed symmetric operator withinfinite deficiency indices while for operator with finitedeficiency indices we prove impossibility of such a kindfactorization. A construction of pairs $\mathcal L_0\subset\mathcal L$ ($\mathcal L_0$ isclosed and densely defined, $\mathcal L=\mathcal L^*\ge 0$) having the property${\rm dom\,}(\mathcal L\mathcal L_0)=\{0\}$ (and, in particular, ${\rm dom\,}(\mathcal L^2_0)=\{0\}$) is given.

Operator-norm approximations of holomorphic one-parameter semigroups of contractions in Hilbert spaces

Yury Arlinskiĭ

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 101-110

We establish the operator-norm convergence of the Iosida and Dunford-Segal approximation formulas for one-parameter semigroups of the class $C_0$, gene ated by maximal sectorial generators in separable Hilbert spaces. Our approach is essentially based on the Crouzeix-Delyon theorem [8] related to the generalization of the von Neumann inequality.

Conservative discrete time-invariant systems and block operator CMV matrices

Yury Arlinskiĭ

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 15 (2009), no. 3, 201-236

It is well known that an operator-valued function $\Theta$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as a transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $\Theta$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so-called "Schur para-me ers". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are analogs of Cantero--Moral--Vel\'azquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

Some remarks on singular perturbations of self-adjoint operators

Yu. M. Arlinskiĭ, E. R. Tsekanovskiĭ

Methods Funct. Anal. Topology 9 (2003), no. 4, 287-308

On $M$-accretive extensions and restrictions

Yu. M. Arlinskii

Methods Funct. Anal. Topology 4 (1998), no. 3, 1-26

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