Related papers: Functions of compact operators under trace class p…
We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier,…
A first order trace formula is obtained for a higher-order differential operator on a segment in the case where the perturbation is an operator of multiplication by a finite complex-valued measure. For the operators of even order $n\ge4$ a…
We study discrete Schroedinger operators with compactly supported potentials on the square lattice. Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely…
The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in…
We study the set of possible traces of anisotropic least gradient functions. We show that even on the unit disk it changes with the anisotropic norm: for two sufficiently regular strictly convex norms the trace spaces coincide if and only…
We present conditions for a family $\left(A\left(x\right)\right)_{x\in\mathbb{R}^{d}}$ of self-adjoint operators in $H^{r}=\mathbb{C}^{r}\otimes H$ for a separable complex Hilbert space $H$, such that the Callias operator…
We study trace functions on the form $ t\to\tr f(A+tB) $ where $ f $ is a real function defined on the positive half-line, and $ A $ and $ B $ are matrices such that $ A $ is positive definite and $ B $ is positive semi-definite. If $ f $…
In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its…
We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous operator $T: Y \longrightarrow X$, we prove that $T$ is a limited operator if and only if,…
This is a continuation of our papers \cite{AP2} and \cite{AP3}. In those papers we obtained estimates for finite differences $(\D_Kf)(A)=f(A+K)-f(A)$ of the order 1 and $(\D_K^mf)(A)\df\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$ of…
We prove that if A and B are bounded self-adjoint operators such that A-B belongs to the trace class, then |A| -|B| belongs to the principal ideal L_{1,\infty} in the algebra L(H) of all bounded operators on an infinite-dimensional Hilbert…
Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is…
In this paper, the definition of noncommutative Orlicz sequence spaces is given, these spaces generalize the Schatten classes Sp(H). After some relations of trace and norm on this spaces have been researched, one give the criterion of…
In this paper we continue the study of groups of trace class and consider in particular the case of semi-direct products. One of the highlights is the theorem saying that the semi-direct product of a semisimple Lie group G and its Lie…
We establish sharp trace- and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov…
We study compactness along orthonormal (disjoint bounded) sequences for operators from Hilbert spaces (Banach lattices) to Frechet spaces.
Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $\omega$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_{\omega}(K)$ if…
We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy Lipschitz condition of the order $0<\a\le 1.$ We prove that if for a function $f$ the $\operatorname{Lip} \a-$ norms of these sections…
Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $\varphi\mapsto\varphi(A,B)$ for functions $\varphi$ in the Besov class…
For an operator monotone function $f(t)$ on the positive real line, we show the operator monotonicity of the type of the functions $(t-a)(t-b)/(f(t)-f(a))(f^\sharp(t)-f^\sharp(b))$.