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Monday, July 27, 2020 | History

2 edition of An inequality for self-adjoint operators on a Hilbert space found in the catalog.

# An inequality for self-adjoint operators on a Hilbert space

## by Herbert J. Bernstein

Published by Courant Institute of Mathematical Sciences, New York University in New York .
Written in English

The Physical Object
Pagination5 p.
ID Numbers
Open LibraryOL25400585M

On the other hand, it is shown in that if f: J → R is an operator convex function on the interval J, then for any self-adjoint operators S and T with spectra in J we have the inequality f (S + T 2) ≤ ∫ 0 1 f ((1 − t) S + t T) d t ≤ f (S) + f (T) 2. according algebra apply assume asymptotic basis belongs Boolean Borel bounded called channel Chapter Chapter IV closed commute complete concept Consequently consider containing continuous converges corresponding defined Definition denote dense derive differential domain easily element energy equal equation equivalence establish Euclidean space.

Some reverses of the Jensen inequality for functions of self-adjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also by:   He also presented some Chebyshev inequalities for self-adjoint operators acting on a Hilbert space in. In this paper we provide several operator extensions of the Chebyshev inequality. In the second section, we present our main results dealing with the Hadamard product of Hilbert space operators and weighted operator geometric by: 5.

However, since the spectrum of a a self-adjoint operator is real the theorem should be true in real Hilbert spaces. I can imagine an argument by complexification but there a . Definition. Let be a Hilbert space and () be the set of bounded operators , an operator ∈ is said to be a compact operator if the image of each bounded set under is relatively compact.. Some general properties. We list in this section some general properties of compact operators. If X and Y are Hilbert spaces (in fact X Banach and Y normed will suffice), then T: X → Y is compact.

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### An inequality for self-adjoint operators on a Hilbert space by Herbert J. Bernstein Download PDF EPUB FB2

Mond-Pecaric Method in Operator Inequalities (Inequalities for bounded selfadjoint operators on a Hilbert space) (Monographs in Inequalities) Hardcover – January 1, by Takayuki Furuta (Author), Jadranka Micic Hot (Author), Josip Pecaric (Author), Yuki Seo (Author) & 1 moreCited by: An inequality is proved in abstract separable Hilbert space H where A and B are bounded self-adjoint positive operators defined in H such that R(A)=R(B) and R(A) is closed.

View full-text Article. Even more, the concept of convex function of operators, its properties including inequalities and applications, defined for continuous functions on bounded self-adjoint operators in Hilbert space.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators.

Furuta, J. Mićić Hot, J. Pečarić, Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities for bounded self adjoint operators on a Hilbert space, element, Zagreb () Google Scholar.

inequality. We will show that under some assumptions this relation is anti-symmetric. Introduction Let f(t) be a continuous, increasing concave function on the real line R and let A be a bounded self-adjoint operator on some Hilbert space H with an in-ner product h,i.

Then for each unit vector ξ ∈ H, we have so-called Jensen inequality. for any and. Since in the operator order, then which gives that, that is, for any, which implies that for ore, which together with () prove the last part of ().

The case of Lipschitzian functions is as follows. Theorem Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers, and let be its spectral by: 2.

Statement of the inequality. The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that |, | ≤, ⋅, where ⋅, ⋅ is the inner es of inner products include the real and complex dot product; see the examples in inner lently, by taking the square root of both sides, and referring to the norms of the vectors, the.

Some norm inequalities for weighted power means of Hilbert space operators are proved for the general class of unitarily invariant norms.

These inequalities generalize a recent inequality of : Danko Jocic. The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators Reviews: 1.

In a similar way given a Hilbert space H, L(H) will be the algebra of all linear bounded operators on H, L sa(H) the real vector space of self-adjoint operators, and L(H)+ the cone of all positive operators.

For every C∈ L(H) its range will be denoted by R(C), its null space will. sa denotes the convex set of bounded self-adjoint operators on the Hilbert space H with spectra in a real interval I. A partial order is naturally equipped on B(H)sa by deﬁning A ≤ B if and only if B − A ∈ B(H)+.

We write A > 0 to mean that A is a strictly positive operator, or equivalently, A ≥ 0 and A is invertible. operator L deﬁned in equation() is self-adjoint.

The following result gives a useful condition for telling when an operator on a complex Hilbert space is self-adjoint. Proposition Let H be a complex Hilbert space (i.e., F = C), and let A ∈ B(H) be given. Then: A is self-adjoint ⇐⇒ hAf,fi ∈ R ∀f ∈ H.

\$\begingroup\$ I think you forgot to write self-adjoint in your first sentence, even though you write self-adjoint in the title.

\$\endgroup\$ – DisintegratingByParts Sep 9. Spectral Theory of Self-Adjoint Operators in Hilbert Space (Mathematics and its Applications) Hardcover – by Michael Sh. Birman (Author), M.Z. Solomjak (Author) See all 3 formats and editions Hide other formats and editions.

Price New from Used from Cited by:   This paper conducts a further study to the development of the existing theory of Jensen’s inequality for self-adjoint operators in a Hilbert space.

The main contribution is the obtained complementary to Jensen’s inequality for general real-valued twice differentiable functions. In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N.

Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are. Hardy-Hilbert's inequality and power inequalities for Berezin numbers of operators Author: Mubariz T.

Garayev, Mehmet Gürdal and Arzu Okudan Subject: Math. Inequal. Appl., 19, 3 () Keywords: 47A63, Hardy inequality, Hardy-Hilbert inequality, Berezin symbol, Berezin number, positive operator, self-adjoint operator Created Date.

The operator A∗ is called the adjoint operator of A. If A = A∗, we say that A is self-adjoint. By the deﬁnition of A∗ we have that the self-adjoint operators on a real ﬁnite dimensional Hilbert space are precisely those operators that are represented by symmetric matrices w.r.t.

an. Operators on Hilbert Space Topics to be covered • Operators on Hilbert spaces Special families of operators: adjoints, projections, Hermitian, unitaries, partial isometries, polar decomposition Density matrices and trace class operators B(H) as dual of trace class • Spectral Theory Spectrum and resolvent.The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.A Hilbert space is an abstract vector space possessing the structure of an inner product that allows.Theorem A self-adjoint operator in a rigged Hilbert space has a complete system of generalized eigenvectors, corresponding to real generalized eigenvalues.

This is Theorem 5' in Subsection I of Volume 4 of I. M. Gelfand's Generalized Functions (on pg. ). They define generalized eigenvectors and eigenvalues as follows.