V. I. Mogilevskii

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Articles: 11

Spectral and pseudospectral functions of Hamiltonian systems: development of the results by Arov-Dym and Sakhnovich

Vadim Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 21 (2015), no. 4, 370-402

The main object of the paper is a Hamiltonian system $J y'-B(t)y=\lambda\Delta(t) y$ defined on an interval $[a,b) $ with the regular endpoint $a$. We define a pseudo\-spectral function of a singular system as a matrix-valued distribution function such that the generalized Fourier transform is a partial isometry with the minimally possible kernel. Moreover, we parameterize all spectral and pseudospectral functions of a given system by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov-Dym and Sakhnovich in this direction.

Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation

Vadim Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 21 (2015), no. 1, 76-98

Let $H$ be a finite dimensional Hilbert space and let $[H]$ be the set of all li ear operators in $H$. We consider first-order symmetric system $J y'-B(t)y=\Lambda(t) f(t)$ with $[H]$-valued coefficients defined on an interval $[a,b) $ with the regular endpoint $a$. It is assumed that the corresponding minimal relation $T_{\rm min}$ has maximally possible deficiency index $n_+(T_{\rm min})=\dim H$. The main result is a parametrization of all characteristic matrices and pseudospectral (spectral) functions of a given system by means of a Nevanlinna type boundary parameter $\tau$. Similar parametrization for regular systems has earlier been obtained by Langer and Textorius. We also show that the coefficients of the parametrization form the matrix $W(\lambda)$ with the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.

Erratum: "On generalized resolvents and characteristic matrices of first-order symmetric systems'', MFAT, Vol. 20, No. 4, 2014, pp. 328–348

Vadim Mogilevskii

Methods Funct. Anal. Topology 21 (2015), no. 1, 100-100

On generalized rezolvents and characteristic matrices of first-order symmetric systems

Tim Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 20 (2014), no. 4, 328-348

We study general (not necessarily Hamiltonian) first-ordersymmetric system $J y'-B(t)y=\Delta(t) f(t)$ on an interval $\mathcal I=[a,b)$ with the regular endpoint $a$ and singular endpoint $b$. It isassumed that the deficiency indices $n_\pm(T_{\min})$ of thecorresponding minimal relation $T_{\min}$ in $L_\Delta^2(\mathcal I)$ satisfy$n_-(T_{\min})\leq n_+(T_{\min})$. We describe all generalized resolvents$y=R(\lambda)f, \; f\in L_\Delta^2(\mathcal I),$ of $T_{\min}$ in terms of boundary problemswith $\lambda$-depending boundary conditions imposed on regular andsingular boundary values of a function $y$ at the endpoints $a$and $b$ respectively. We also parametrize all characteristicmatrices $\Omega(\lambda)$ of the system immediately in terms of boundaryconditions. Such a parametrization is given both by the blockrepresentation of $\Omega(\lambda)$ and by the formula similar to thewell-known Krein formula for resolvents. These results develop the Straus' results on generalized resolvents and characteristicmatrices of differential operators.

On exit space extensions of symmetric operators with applications to first order symmetric systems

V. I. Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 19 (2013), no. 3, 268-292

Let $A$ be a symmetric linear relation with arbitrary deficiency indices. By using the conceptof the boundary triplet we describe exit space self-adjointextensions $\widetilde A^\tau$ of $A$ in terms of a boundary parameter $\tau$. We characterize certain geometrical properties of $\widetilde A^\tau$ and describe all $\widetilde A^\tau$ with ${\rm mul}\, \widetilde A^\tau=\{0\}$. Applying these results to general (possibly non-Hamiltonian) symmetric systems $Jy'- B(t)y=\Delta(t)y, \; t \in [a,b\rangle,$ we describe all matrix spectral functions of theminimally possible dimension such that the Parseval equality holdsfor any function $f\in L_\Delta^2([a,b \rangle)$.

Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices

Vadim Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 15 (2009), no. 3, 280-300

Let $l [y]$ be a formally selfadjoint differential expression of an even order on the interval $[0,b \rangle$, $b\leq \infty$, with operator coefficients, acting in a separable Hilbert space $H$. We introduce the concept of deficiency indices $n_{b\pm}$ of the expression $l$ at the point $b$ and show that in the case $\dim H=\infty$ any values of $n_{b\pm}$ are possible. Moreover the decomposing selfadjoint boundary conditions exist if and only if $n_{b+}=n_{b-}$. Our considerations of differential operators with arbitrary (possibly unequal) deficiency indices are based on the concept of a decomposing $D$-boundary triplet. Such an approach enables to describe extensions of the minimal operator directly in terms of operator boundary conditions at the ends of the interval $[0,b \rangle$. In particular we describe in a compact form selfadjoint decomposing boundary conditions.

Associated to a $D$-triplet is an $m$-function, which can be regarded as a gene alization of the classical characteristic (Titchmarsh-Weyl) function. Our definition enables to describe all $m$-functions (and, therefore, all spectral functions) directly in terms of boundary conditions at the right end $b$.

Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers

Vadim Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 12 (2006), no. 3, 258-280

Let $H$ be a Hilbert space and let $A$ be a symmetric operator in $H$ with arbitrary (not necessarily equal) deficiency indices $n_\pm (A)$. We introduce a new concept of a $D$-boundary triplet for $A^*$, which may be considered as a natural generalization of the known concept of a boundary triplet (boundary value space) for an operator with equal deficiency indices. With a $D$-triplet for $A^*$ we associate two Weyl functions $M_+(\cdot)$ and $M_-(\cdot)$. It is proved that the functions $M_\pm(\cdot)$ posses a number of properties similar to those of the known Weyl functions ($Q$-functions) for the case $n_+(A)=n_-(A)$. We show that every $D$-triplet for $A^*$ gives rise to Krein type formulas for generalized resolvents of the operator $A$ with arbitrary deficiency indices. The resolvent formulas describe the set of all generalized resolvents by means of two pairs of operator functions which belongs to the Nevanlinna type class $\bar R(H_0,H_1)$. This class has been earlier introduced by the author.

Nevanlinna type families of linear relations and the dilation theorem

Vadim Mogilevskii

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 12 (2006), no. 1, 38-56

Let H1 be a subspace in a Hilbert space H0 and let $\widetilde C(H_0,H_1)$ be the set of all closed linear relations from $H_0$ to $H_1$. We introduce a Nevanlinna type class $\widetilde R_+ (H_0,H_1)$ of holomorphic functions with values in $\widetilde C(H_0,H_1)$ and investigate its properties. In particular we prove the existence of a dilation for every function $\tau_+(\cdot)\in \widetilde R_+ (H_0,H_1)$. In what follows these results will be used for the derivation of the Krein type formula for generalized resolvents of a symmetric operator with arbitrary (not necessarily equal) deficiency indices.

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

Krein type formula for canonical resolvents of dual pairs of linear relations

M. M. Malamud, V. I. Mogilevskii

Methods Funct. Anal. Topology 8 (2002), no. 4, 72-100

Weak generalized limit and ergodic theorems

Vadim Mogilevskii, Aleksander Revenko

Methods Funct. Anal. Topology 7 (2001), no. 2, 52-67


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