Related papers: An Operator-Fractal
The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,\mu)$,…
A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators,…
Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic…
Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of…
Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$ Lf(x) = \frac12 {\rm Tr} Q D^2 f(x) + <Ax, Df(x)>.$$ Here $Q$ is a positive symmetric operator from $E^*$ to…
By the help of power series f we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f. Utilising these functions we prove some inequalities for the spectral radius of the bounded…
The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable…
We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is…
This thesis is devoted to the study of multivariate (joint) spectral multipliers for systems of strongly commuting non-negative self-adjoint operators, $L=(L_1,\ldots,L_d),$ on $L^2(X,\nu),$ where $(X,\nu)$ is a measure space. By strong…
We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several deterministic and almost sure…
In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e., $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a…
We explore spectral duality in the context of measures in $\br^n$, starting with partial differential operators and Fuglede's question (1974) about the relationship between orthogonal bases of complex exponentials in $L^2(\Omega)$ and…
In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators $$\mathcal{L}v(x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle+Df(v_{\star}(x))v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2$$ where…
We introduce the concept of a \mu-scale invariant operator with respect to unitary transformation in a separable complex Hilbert space. We show that if a nonnegative densely defined symmetric operator is \mu-scale invariant for some \mu >0,…
Extending the notion of bounded variation, a function $u \in L_c^1(\mathbb R^n)$ is of bounded fractional variation with respect to some exponent $\alpha$ if there is a finite constant $C \geq 0$ such that the estimate \[ \biggl|\int u(x)…
Let $\mu$ denot the infinite convolution generated by $\{(N_k,B_k)\}_{k=1}^\infty$ given by $$ \mu =\delta_{{N_1}^{-1}B_1}\ast\delta_{(N_1N_2)^{-1}B_2}\ast\dots\ast\delta_{(N_1N_2\cdots N_k)^{-1}B_k} *\cdots. $$ where $B_k$ is a complete…
The aim of this paper is to prove new uncertainty principles for an integral operator $\tt$ with a bounded kernel for which there is a Plancherel theorem. The first of these results is an extension of Faris's local uncertainty principle…
A network is a countable, connected graph X viewed as a one-complex, where each edge [x,y]=[y,x] (x,y in X^0, the vertex set) is a copy of the unit interval within the graph's one-skeleton X^1 and is assigned a positive conductance c(xy). A…
Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of…
Spectral decomposition with respect to the wave functions of Ruijsenaars hyperbolic system defines an integral transform, which generalizes classical Fourier integral. For a certain class of analytical symmetric functions we prove inversion…