arXiv subjects

Yury Arlinski\uı

Publications and source records attributed to Yury Arlinski\uı.

3 recordsLinked to original sources

arXiv2021

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

In this paper holomorphic families of linear relations which 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 closed forms associated with the Stieltjes and inverse Stieltjes families of linear relations are holomorphic families 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.

math.FA↗

arXiv2018

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.

math.FA↗

arXiv2018

Stieltjes and inverse Stieltjes holomorphic families of linear relations and their representations

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.

math.FA↗