706 research outputs found
Q-functions and boundary triplets of nonnegative operators
Operator-valued -functions for special pairs of nonnegative selfadjoint
extensions of nonnegative not necessarily densely defined operators are defined
and their analytical properties are studied. It is shown that the Kre\u\i
n-Ovcharenko statement announced in \cite{KrO2} is valid only for -functions
of densely defined symmetric operators with finite deficiency indices. A
general class of boundary triplets for a densely defined nonnegative operator
is constructed such that the corresponding Weyl functions are of Kre\u\i
n-Ovcharenko type
On J-Self-Adjoint Operators with Stable C-Symmetry
The paper is devoted to a development of the theory of self-adjoint operators
in Krein spaces (J-self-adjoint operators) involving some additional properties
arising from the existence of C-symmetries. The main attention is paid to the
recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric
operator S. The general results are specialized further by studying in detail
the case where S has defect numbers
Completion, extension, factorization, and lifting of operators with a negative index
The famous results of M.G. Kre\u{\i}n concerning the description of
selfadjoint contractive extensions of a Hermitian contraction and the
characterization of all nonnegative selfadjoint extensions of a
nonnegative operator via the inequalities ,
where and are the Kre\u{\i}n-von Neumann extension and the
Friedrichs extension of , are generalized to the situation, where
is allowed to have a fixed number of negative eigenvalues. These
generalizations are shown to be possible under a certain minimality condition
on the negative index of the operators and , respectively;
these conditions are automatically satisfied if is contractive or is
nonnegative, respectively.
The approach developed in this paper starts by establishing first a
generalization of an old result due to Yu.L. Shmul'yan on completions of
nonnegative block operators. The extension of this fundamental
result allows us to prove analogs of the above mentioned results of M.G.
Kre\u{\i}n and, in addition, to solve some related lifting problems for
-contractive operators in Hilbert, Pontryagin and Kre\u{\i}n spaces in a
simple manner. Also some new factorization results are derived, for instance, a
generalization of the well-known Douglas factorization of Hilbert space
operators. In the final steps of the treatment some very recent results
concerning inequalities between semibounded selfadjoint relations and their
inverses turn out to be central in order to treat the ordering of
non-contractive selfadjoint operators under Cayley transforms properly.Comment: 29 page
A general realization theorem for matrix-valued Herglotz-Nevanlinna functions
New special types of stationary conservative impedance and scattering
systems, the so-called non-canonical systems, involving triplets of Hilbert
spaces and projection operators, are considered. It is established that every
matrix-valued Herglotz-Nevanlinna function of the form
V(z)=Q+Lz+\int_{\dR}(\frac{1}{t-z}-\frac{t}{1+t^2})d\Sigma(t) can be realized
as a transfer function of such a new type of conservative impedance system. In
this case it is shown that the realization can be chosen such that the main and
the projection operators of the realizing system satisfy a certain
commutativity condition if and only if L=0. It is also shown that with
an additional condition (namely, is invertible or L=0), can be realized as
a linear fractional transformation of the transfer function of a non-canonical
scattering -system. In particular, this means that every scalar
Herglotz-Nevanlinna function can be realized in the above sense.
Moreover, the classical Livsic systems (Brodskii-Livsic operator
colligations) can be derived from -systems as a special case when
and the spectral measure is compactly supported. The realization
theorems proved in this paper are strongly connected with, and complement the
recent results by Ball and Staffans.Comment: 28 page
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