Variation of the Jordan structure of G-self-adjoint operators and self-adjoint operator functions under small perturbations

1989 ◽  
Vol 22 (3) ◽  
pp. 236-237
Author(s):  
V. R. Ol'shevskii
Keyword(s):  
Adjoint Operator ◽  
Small Perturbations ◽  
Jordan Structure ◽  
Operator Functions ◽  
Adjoint Operators

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.

On the essential spectrum of self-adjoint operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 261-274
Author(s):  
B. J. Harris
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Adjoint Operator ◽  
Matrix Differential ◽  
Image Position ◽  
Adjoint Operators

SynopsisWe provide estimates of the formfor the length of gap centre μ in the essential spectrum of a self-adjoint operator generated by a matrix differential expression.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals HILBERT–PÓLYA CONJECTURE, ZETA FUNCTIONS AND BOSONIC QUANTUM FIELD THEORIES

2013 ◽  
Vol 28 (17) ◽  
pp. 1350072 ◽  
Author(s):  
JULIO C. ANDRADE
Keyword(s):  
Adjoint Operator ◽  
Zeta Functions ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Functional Integrals ◽  
Generating Functional ◽  
Nontrivial Zeros ◽  
Riemann Zeta ◽  
Adjoint Operators

The original Hilbert and Pólya conjecture is the assertion that the nontrivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However, the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by Number Theory. In this paper, we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularizations, is that the various Dirichlet series attached to such sequences — such as those which are sums over "primes" of ( norm P)-s have a natural boundary, i.e. they cannot be continued beyond the line Re (s) = 0. The main argument is that once the regularized determinant of a Laplacian is meromorphic in s, it follows that the series considered above cannot be a regularized determinant. In other words, we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed.

scholarly journals The square root of a positive self-adjoint operator

1968 ◽  
Vol 8 (1) ◽  
pp. 17-36 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Elementary Proof ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Spectral Theorem ◽  
Square Root ◽  
Spaces Of Continuous Functions ◽  
Elementary Construction ◽  
Adjoint Operators ◽  
Spectral Family ◽  
Insight Into

One elementary proof of the spectral theorem for bounded self-adjoint operators depends on an elementary construction for the square root of a bounded positive self-adjoint operator. The purpose of this paper is to give an elementary construction for the unbounded case and to deduce the spectral theorem for unbounded self-adjoint operators. In so far as all our results are more or less immediate consequences of the spectral theorem there is little is entirely new. On the other hand the elementary approach seems to the author to provide a deeper insight into the structure of the problem and also leads directly to the spectral theorem without appealing first to the bounded case. Besides this, our methods for proving uniqueness of the square root and of the spectral family seem to be new even in the bounded case. In particular there is no need to invoke representation theorems for linear functionals on spaces of continuous functions.

Self-adjoint operators on inner product spaces over fields of power series

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Hans Keller ◽  
Ochsenius Herminia
Keyword(s):  
Power Series ◽  
Adjoint Operator ◽  
Inner Product ◽  
Closed Field ◽  
Product Spaces ◽  
Binary Forms ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Orthogonal Decompositions ◽  
Adjoint Operators

AbstractTheorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on ℝn. In the paper we consider finite dimensional inner product spaces (E, ϕ) over a field K = F((χ 1, ..., x m)) of generalized power series in m variables and with coefficients in a real closed field F. It turns out that for most of these spaces (E, ϕ) every self-adjoint operator gives rise to an orthogonal decomposition of E into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator T: (E, ϕ) → (E, ϕ) is decomposable except when dim E is a power of 2 with exponent at most m, and ϕ is a tensor product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators.

A fixed point method to solve linear operator equations involving self-adjoint operators in Hilbert space

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3249-3251
Author(s):  
Mohammad Khan ◽  
Dinu Teodorescu
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Fixed Point Method ◽  
Operator Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Adjoint Operators ◽  
Invertible Matrices ◽  
Linear Operator Equations

In this paper we provide existence and uniqueness results for linear operator equations of the form (I+Am) x = f , where A is a self-adjoint operator in Hilbert space. Some applications to the study of invertible matrices are also presented.

Preconditioned acoustic least-squares two-way wave-equation migration with exact adjoint operator

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. S1-S13 ◽  
Author(s):  
Linan Xu ◽  
Mauricio D. Sacchi
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Adjoint Operator ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Product Test ◽  
Dot Product ◽  
The Time Domain ◽  
Adjoint Operators

We have investigated the problem of designing the forward operator and its exact adjoint for two-way wave-equation least-squares migration. We study the problem in the time domain and pay particular attention to the individual operators that are required by the algorithm. We derive our algorithm using the language of linear algebra and establish a simple path to design forward and adjoint operators that pass the dot-product test. We also found that the exact adjoint operator is not equal to the classic reverse time migration algorithm. For instance, one must pay particular attention to boundary conditions to compute the exact adjoint that accurately passes the dot-product test. Forward and adjoint operators are adopted to solve the so-called least-squares reverse time migration problem via the method of conjugate gradients. We also examine a preconditioning strategy to invert extended images.

scholarly journals Spectral Multipliers of Self-Adjoint Operators on Besov and Triebel–Lizorkin Spaces Associated to Operators

2019 ◽  
Author(s):  
The Anh Bui ◽  
Xuan Thinh Duong
Keyword(s):  
Heat Kernel ◽  
Hardy Spaces ◽  
Adjoint Operator ◽  
Space Of Homogeneous Type ◽  
Multiplier Theorem ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Gaussian Estimate ◽  
Adjoint Operators

Abstract Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

scholarly journals Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

1998 ◽  
Vol 1 ◽  
pp. 42-74 ◽  
Author(s):  
E.B. Davies
Keyword(s):  
Spectral Properties ◽  
Linear Subspace ◽  
Adjoint Operator ◽  
Dimensional Linear Subspace ◽  
Finite Dimensional ◽  
Adjoint Operators ◽  
Computation Of Eigenvalues

AbstractThis paper considers a number of related problems concerning the computation of eigenvalues and complex resonances of a general self-adjoint operator H. The feature which ties the different sections together is that one restricts oneself to spectral properties of H which can be proved by using only vectors from a pre-assigned (possibly finite-dimensional) linear subspace L.

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