Estimates of the norm of the resolvent of a non-self-adjoint operator which is ?nearly? self-adjoint or unitary

1983 ◽  
Vol 33 (2) ◽  
pp. 81-84 ◽  
Author(s):  
M. I. Gil'
Keyword(s):  
Adjoint Operator ◽  
Norm Of The Resolvent

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations

2020 ◽  
Vol 18 (1) ◽  
pp. 1615-1624
Author(s):  
Guangyu An ◽  
Ying Yao
Keyword(s):  
Operator Algebras ◽  
Adjoint Operator ◽  
Rassias Stability

Abstract In this paper, we study the Hyers-Ulam-Rassias stability of ( m , n ) (m,n) -Jordan derivations. As applications, we characterize ( m , n ) (m,n) -Jordan derivations on C ⁎ {C}^{\ast } -algebras and some non-self-adjoint operator algebras.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

2000 ◽  
Vol 11 (08) ◽  
pp. 1057-1078
Author(s):  
JINGBO XIA
Keyword(s):  
Hausdorff Measure ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Trace Class ◽  
Multiplication Operators ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS

2021 ◽  
Vol 9 (1) ◽  
pp. 107-127
Author(s):  
R. Kolisnyk ◽  
V. Gorodetskyi ◽  
O. Martynyuk
Keyword(s):  
Fundamental Solution ◽  
Initial Function ◽  
Adjoint Operator ◽  
Differentiation Operator ◽  
The Self ◽  
Multipoint Problem ◽  
Time Problem ◽  
Sigma 1 ◽  
Non Local ◽  
Differential Operator Equation

In this paper we investigate the differential-operator equation $$ \partial u (t, x) / \partial t + \varphi (i \partial / \partial x) u (t, x) = 0, \quad (t, x) \in (0, + \infty) \times \mathbb {R} \equiv \Omega, $$ where the function $ \varphi \in C ^ {\infty} (\mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i \partial / \partial x $, in $ L_2 (\mathbb {R}) $ it is established that the operator $ \varphi (i \partial / \partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ \partial u / \partial t + \sqrt {I- \Delta} u = 0 $, $ \Delta = D_x ^ 2 $, with the fractionation differentiation operator $ \sqrt { I- \Delta} = \varphi (i \partial / \partial x) $, where $ \varphi (\sigma) = (1+ \sigma ^ 2) ^ {1/2} $, $ \sigma \in \mathbb {R} $ is attributed to the considered equation. Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, \cdot) $ for $ t \to + \infty $ (solution stabilization) in spaces of type $ S '$.

scholarly journals LOCALIZATION OF THE NUMBER OF PHOTONS OF GROUND STATES IN NONRELATIVISTIC QED

2003 ◽  
Vol 15 (03) ◽  
pp. 271-312 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Radiation Field ◽  
Ground State ◽  
Fock Space ◽  
Adjoint Operator ◽  
Ground States ◽  
Electron System ◽  
Number Operator ◽  
Position Variable ◽  
Boson Fock Space ◽  
Quantized Radiation Field

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2 (ℝ3) ⊗ ℱ ≅ L2 (ℝ3; ℱ), where ℱ is the Boson Fock space over L2 (ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to [Formula: see text], where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ‖(1 ⊗ Nk/2) ψg (x)‖ℱ ≤ Dk e-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular [Formula: see text] for 0 < β < δ/2 is obtained.

scholarly journals ON EQUALITY CONDITION FOR TRACE JENSEN INEQUALITY IN SEMI-FINITE VON NEUMANN ALGEBRAS

2008 ◽  
Vol 19 (04) ◽  
pp. 481-501 ◽  
Author(s):  
TETSUO HARADA ◽  
HIDEKI KOSAKI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Strict Convexity ◽  
Jensen Inequality ◽  
Von Neumann ◽  
Convexity Assumption ◽  
Finite Von Neumann Algebra ◽  
Equality Condition ◽  
Finite Von Neumann Algebras

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

scholarly journals Recovery of inclusion separations in strongly heterogeneous composites from effective property measurements

2011 ◽  
Vol 468 (2139) ◽  
pp. 784-809 ◽  
Author(s):  
Chris Orum ◽  
Elena Cherkaev ◽  
Kenneth M. Golden
Keyword(s):  
Inverse Spectral Problem ◽  
Algebraic Curves ◽  
Volume Fraction ◽  
Adjoint Operator ◽  
Effective Property ◽  
Host Matrix ◽  
Separation Parameter ◽  
Property Data ◽  
Transformation Structure ◽  
Parameter Values

An effective property of a composite material consisting of inclusions within a host matrix depends on the geometry and connectedness of the inclusions. This dependence may be quite strong if the constituents have highly contrasting properties. Here, we consider the inverse problem of using effective property data to obtain information on the geometry of the microstructure. While previous work has been devoted to recovering the volume fractions of the constituents, our focus is on their connectedness—a key feature in critical behaviour and phase transitions. We solve exactly a reduced inverse spectral problem by bounding the volume fraction of the constituents, an inclusion separation parameter and the spectral gap of a self-adjoint operator that depends on the geometry of the composite. We present a new algorithm based on the Möbius transformation structure of the forward bounds whose output is a set of algebraic curves in parameter space bounding regions of admissible parameter values. These results advance the development of techniques for characterizing the microstructure of composite materials. As an example, we obtain inverse bounds on the volume fraction and separation of the brine inclusions in sea ice from measurements of its effective complex permittivity.

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