S. Hassi

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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.

Selfadjoint extensions of relations whose domain and range are orthogonal

S. Hassi, J.-Ph. Labrousse, H.S.V. de Snoo

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 26 (2020), no. 1, 39-62

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space $\mathfrak H_1$ to a Hilbert space $\mathfrak H_2$ are considered in the Hilbert space $\mathfrak H_1\oplus\mathfrak H_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in $\mathfrak H_1 \oplus \mathfrak H_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative selfadjoint extensions $A$, in particular the Friedrichs and Krein-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that $A$ belongs to the class of extremal extensions of $S$ if and only if ${\rm dom\,} A \perp {\rm ran\,} A$. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of $S$ and semibounded parameters describing them if and only if the operator part of $R$ is bounded.

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

Yu. M. Arlinskiĭ, S. Hassi

↓ Abstract   |   Article (.pdf)

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.

Generalized zeros and poles of $\mathcal N_\kappa$-functions: on the underlying spectral structure

Seppo Hassi, Annemarie Luger

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 12 (2006), no. 2, 131-150

Let $q$ be a scalar generalized Nevanlinna function, $q\in\mathcal N_\kappa$. Its gene alized zeros and poles (including their orders) are defined in terms of the function's operator representation. In this paper analytic properties associated with the underlying root subspaces and their geometric structures are investigated in terms of the local behaviour of the function. The main results and various characterizations are expressed by means of (local) moments, asymptotic expansions, and via the basic factorization of $q$. Also an inverse problem for recovering the geometric structure of the root subspace from an appropriate asymptotic expansion is solved.

Generalized resolvents and boundary triplets for dual pairs of linear relations

Seppo Hassi, Mark Malamud, Vadim Mogilevskii

Methods Funct. Anal. Topology 11 (2005), no. 2, 170-187

Operator models associated with singular perturbations

Henk de Snoo, Vladimir Derkach, Seppo Hassi

Methods Funct. Anal. Topology 7 (2001), no. 3, 1-21

Generalized resolvents of symmetric operators and admissibility

H. S. V. de Snoo, V. A. Derkach, S. Hassi, M. M. Malamud

Methods Funct. Anal. Topology 6 (2000), no. 3, 24-55

Operator models associated with Kac subclasses of generalized Nevanlinna functions

Henk de Snoo, Vladimir Derkach, Seppo Hassi

Methods Funct. Anal. Topology 5 (1999), no. 1, 65-87


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