# Vol. 12 (2006), no. 3

### Higher powers of q-deformed white noise

Methods Funct. Anal. Topology **12** (2006), no. 3, 205-219

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.

### Borg-type theorems for generalized Jacobi matrices and trace formulas

Methods Funct. Anal. Topology **12** (2006), no. 3, 220-233

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.

### $\nabla$-Fredgolm operators in Banach-Kantorovich spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 234-242

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 self-adjoint perturbations

Methods Funct. Anal. Topology **12** (2006), no. 3, 243-253

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.

### A note on one decomposition of Banach spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 254-257

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.

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

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.

### A locally convex quotient cone

Asghar Ranjbari, Husain Saiflu

Methods Funct. Anal. Topology **12** (2006), no. 3, 281-285

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

### On similarity of convolution Volterra operators in Sobolev spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 286-295

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.

### On enveloping $C^*$-algebra of one affine Temperley-Lieb algebra

Methods Funct. Anal. Topology **12** (2006), no. 3, 296-300

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