scholarly journals A characterization of singular packing subspaces with an application to limit-periodic operators

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Silas L. Carvalho ◽  
César R. de Oliveira
Keyword(s):  
Packing Dimension ◽  
Spectral Measures ◽  
Adjoint Operators ◽  
Periodic Operators

AbstractA new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a

Monotone convergence theorems for semi-bounded operators and forms with applications

2010 ◽  
Vol 140 (5) ◽  
pp. 927-951 ◽  
Author(s):  
Jussi Behrndt ◽  
Seppo Hassi ◽  
Henk de Snoo ◽  
Rudi Wietsma
Keyword(s):  
Differential Operators ◽  
Multiplication Operators ◽  
Monotone Convergence ◽  
Monotone Sequence ◽  
Bounded Operators ◽  
Von Neumann ◽  
Sturm Liouville ◽  
Adjoint Operators ◽  
Liouville Operators

AbstractLet Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

scholarly journals On spectra and Brown's spectral measures of elements in free products of matrix algebras

2008 ◽  
Vol 103 (1) ◽  
pp. 77
Author(s):  
Junsheng Fang ◽  
Don Hadwin ◽  
Xiujuan Ma
Keyword(s):  
Free Product ◽  
Von Neumann Algebra ◽  
Single Point ◽  
Diagonal Operator ◽  
Spectral Measures ◽  
Free Products ◽  
Hyperinvariant Subspace ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Adjoint Operators

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})$, respectively. We prove that $AB$ is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0$. We show that if $X=AB$ or $X=A+B$ and $A,B$ are not scalar matrices, then the Brown measure of $X$ is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if $X=AB$ or $X=A+B$ and $X\neq \lambda 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$.

scholarly journals A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 385-404 ◽  
Author(s):  
MOHAMED EL-GEBEILY ◽  
DONAL O'REGAN
Keyword(s):  
Complete Characterization ◽  
Second Order ◽  
Inner Product ◽  
Type I ◽  
Weak Formulation ◽  
Friedrichs Extension ◽  
Sesquilinear Form ◽  
Adjoint Operators ◽  
Special Case

AbstractIn this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

BASIC PROPERTIES AND CHARACTERIZATION OF STOCHASTICALLY SELF-SIMILAR PROCESSES IN Rd

Fractals ◽  
1999 ◽  
Vol 07 (01) ◽  
pp. 59-78 ◽  
Author(s):  
DANIELE VENEZIANO
Keyword(s):  
Random Variable ◽  
Random Processes ◽  
Affine Transformations ◽  
Spectral Measures ◽  
Self Similarity ◽  
Basic Properties ◽  
Properties And Characterization ◽  
Classical Notion ◽  
Self Similar

The classical notion of self-similarity (ss) for random X(t) as invariance under the group of positive affine transformations {X→ arX, t→rt; ar>0} is extended by allowing ar to be a random variable. The resulting property of "stochastic self-similarity" (sss) is applied to both ordinary and generalized random processes in Rd, d≥1. The class of sss processes seems to correspond to that of multifractal processes (the latter are variously defined in the literature). The spectral measures of ordinary and generalized sss processes are themselves stochastically self-similar. Two characterizations of ss processes by Lamperti are extended to the sss case and several basic properties of ordinary and generalized sss processes are derived.

scholarly journals On the Self-Adjointness of H+A∗+A

2020 ◽  
Vol 23 (4) ◽  
Author(s):  
Andrea Posilicano
Keyword(s):  
Field Theory ◽  
Annihilation Operator ◽  
The Self ◽  
Quantum Field ◽  
Bounded Operators ◽  
Nelson Model ◽  
Adjoint Operators ◽  
Nonperturbative Theory

AbstractLet $H:\text {dom}(H)\subseteq \mathfrak {F}\to \mathfrak {F}$ H : dom ( H ) ⊆ F → F be self-adjoint and let $A:\text {dom}(H)\to \mathfrak {F}$ A : dom ( H ) → F (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations $\widehat H$ H ̂ of the formal Hamiltonian H + A∗ + A with $\text {dom}(H)\cap \text {dom}(\widehat H)=\{0\}$ dom ( H ) ∩ dom ( H ̂ ) = { 0 } . We give an explicit characterization of $\text {dom}(\widehat H)$ dom ( H ̂ ) and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$ ( − H ̂ + z ) − 1 − ( − H + z ) − 1 . Moreover, we consider the problem of the description of $\widehat H$ H ̂ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$ H + A n ∗ + A n + E n , where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.

scholarly journals Analysis of the limiting spectral measure of large random matrices of the separable covariance type

2014 ◽  
Vol 03 (04) ◽  
pp. 1450016 ◽  
Author(s):  
Romain Couillet ◽  
Walid Hachem
Keyword(s):  
Random Matrix ◽  
Spectral Measure ◽  
Statistical Estimation ◽  
Complete Characterization ◽  
Nonnegative Matrices ◽  
Spectral Measures ◽  
Estimation Problems ◽  
Matrix Formula ◽  
Limiting Spectral Measure

Consider the random matrix [Formula: see text] where D and [Formula: see text] are deterministic Hermitian nonnegative matrices with respective dimensions N × N and n × n, and where X is a random matrix with independent and identically distributed centered elements with variance 1/n. Assume that the dimensions N and n grow to infinity at the same pace, and that the spectral measures of D and [Formula: see text] converge as N, n → ∞ towards two probability measures. Then it is known that the spectral measure of ΣΣ* converges towards a probability measure μ characterized by its Stieltjes transform. In this paper, it is shown that μ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as [Formula: see text] near an edge a of its support. In addition, a complete characterization of the support of μ is provided. Aside from its mathematical interest, the analysis underlying these results finds important applications in a certain class of statistical estimation problems.

scholarly journals The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains

2021 ◽  
Vol 93 (2) ◽  
Author(s):  
Robert Denk ◽  
Markus Kunze ◽  
David Ploß
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Regularity Of Solutions ◽  
Lipschitz Domains ◽  
Lipschitz Boundary ◽  
Full Characterization ◽  
Wentzell Boundary Conditions ◽  
Adjoint Operators

AbstractWe investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d with Lipschitz boundary $$\Gamma $$ Γ . More precisely, using form methods, we show that the associated operator on the ground space $$L^2(\Omega )\times L^2(\Gamma )$$ L 2 ( Ω ) × L 2 ( Γ ) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

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