Vol. 12 (2006), no. 3

We introduce the renormalized powers of $q$-deformed white noise, for any $q$ in the open interval $(-1,1)$, and we extend to them the no--go theorem recently proved by Accardi--Boukas--Franz in the Boson case. The surprising fact is that the lower bound ( ef{basicineq}), which defines the obstruction to the positivity of the sesquilinear form, uniquely determined by the renormalized commutation relations, is independent of $q$ in the half-open interval $(-1,1]$, thus including the Boson case. The exceptional value $q=-1$, corresponding to the Fermion case, is dealt with in the last section of the paper where we prove that the argument used to prove the no--go theorem for $q \ne 0$ does not extend to this case.

The paper deals with two types of inverse spectral problems for the class of generalized Jacobi matrices introduced in [9]. Following the scheme proposed in [5], we deduce analogs of the Hochstadt--Lieberman theorem and the Borg theorem. Properties of a Weyl function of the generalized Jacobi matrix are systematically used to prove the uniqueness theorems. Trace formulas for the generalized Jacobi matrix are also derived.

The paper is devoted to studying $\nabla$-Fredholm operators in Banach--Kantorovich spaces over a ring of measurable functions. We show that a bounded linear operator acting in Banach--Kantorovich space is $\nabla$-Fredholm if and only if it can be represented as a sum of an invertible operator and a cyclically compact operator.

Finite rank perturbations of a semi-bounded self-adjoint operator $A$ are studied. Different types of finite rank perturbations (regular, singular, mixed singular) are described from a unique point of view and by the same formula with the help of quasi-boundary value spaces. As an application, a Schr\"{o}dinger operator with nonlocal point interactions is considered.

For a scalar type spectral operator $A$ in complex Banach space $X$, the decomposition of $X$ into the direct sum \begin{equation*} X=\ker A\oplus \overline{R(A)}, \end{equation*} where $\ker A$ is the kernel of $A$ and $\overline{R(A)}$ is the closure of its range $R(A)$ is established.

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.

We define a quotient locally convex cone and verify some topological properties of it. We show that the extra conditions are necessary.

Necessary and sufficient conditions for a convolution Volterra operator to be similar in a Sobolev space to the operator $J^\alpha$ are obtained. A criterion of similarity is obtained as well.

$C^*$-algebras generated by orthogonal projections satisfying relations of Temperley-Lieb type are constructed.