Double Operator Integrals in a Hilbert Space

Бирман, М. Ш.; Solomyak, Michael

doi:10.1007/s00020-003-1157-8

Cited by 124 publications

(78 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The notion of a ( , )-multiplier is in this case closely related to that of double operator integrals introduced and developed by Birman and Solomyak [3,4,5,6] in connection with various problems of Mathematical Physics and in particular of Perturbation Theory. If ( , ℰ) and ( , ℱ) are spectral measures on the Hilbert spaces and , they defined the double operator integral…”

Section: A Bounded Function : ℕ × ℕ → ℂ Is Called a Schur Multiplier mentioning

confidence: 99%

Multidimensional operator multipliers

Juschenko

Todorov

Turowska

2009

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several * -algebras satisfying certain boundedness conditions. In the case of commutative * -algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding * -algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding * -algebras.

show abstract

Section: A Bounded Function : ℕ × ℕ → ℂ Is Called a Schur Multiplier mentioning

confidence: 99%

Multidimensional operator multipliers

Juschenko

Todorov

Turowska

2009

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Some new applications to the Hilbert space case were also found. All this prompted M. B. and M. Z. Solomyak to publish the paper [13] where a general survey of that theory and its applications (in the Hilbert space framework) was given.…”

Section: S Buslaev M Z Solomyak and D R Yafaevmentioning

confidence: 99%

Untitled

2005

St. Petersburg Math. J.

View full text Add to dashboard Cite

Mikhail Shlemovich Birman reached the age of 75 on the 17th January, 2003. An eminent mathematician, he is the author of numerous fundamental results in the general spectral theory and in the spectral theory of differential operators, both ordinary and partial. He contributed much to several other fields of analysis: function space theory, approximation theory, integral operators, etc.A detailed survey of M. B.'s achievements before 1998 can be found in [1 * ] (see also [3 * ]). For this reason, in the present note we restrict ourselves to the description of his results obtained after 1998, referring to earlier publications only when necessary.Nowadays M. B. is as active as ever. Largely, his new results pertain to two fields: 1) the discrete spectrum of a perturbed operator in gaps of the unperturbed one, and 2) absolute continuity of the spectrum of periodic operators of mathematical physics. Before analyzing these results, we would like to emphasize a characteristic feature of M. B.'s entire work: he knows how to look at a specific problem from a bird's-eye view. Rising over individual peculiarities of a concrete question, he states a new and general problem that includes the initial one (and many others) as a particular case. Next he analyzes the general problem in abstract terms. This either automatically leads to a solution of the initial problem, or-in more complicated cases-reveals questions of analytic nature to be answered for that.The bibliography completing this text consists of two parts. In the first part we give references to the papers [1 * , 3 * ] mentioned above and to the list [2 * ] comprising the publications by M. B. over the same period of time. Five publications occurring in [2 * ] are also included in the first part, with the same numbers as in [2 * ]. For various reasons, the corresponding bibliographic data in [2 * ] were not quite complete, and now we give full references. The second part is the full list of subsequent publications by M. B. When making a reference to the first part, we mark the corresponding number with * . Also, in the text the reader may see references of the type [110 * ] that are absent in both parts of the list. Such a reference means the paper number 110 in the earlier list [2 * ]. §1. Spectrum in gaps. Threshold effects Let A be a selfadjoint operator with a gap (λ − , λ + ) in the spectrum. This means that the interval (λ − , λ + ) is free of points of the spectrum, and the edges λ − and λ + belong to the spectrum (or are infinite).Let V be another operator (a "perturbation"); for definiteness, we assume that V ≥ 0. Denote A ± (α) = A ∓ αV . If V is subordinate to A in an appropriate sense, then for any α > 0, the spectrum of A ± (α) in the gap of A either is empty, or consists of at most countably many eigenvalues (with regard to multiplicity), with the only possible accumulation point λ ± . Moreover, as α increases, the eigenvalues of the operator A + (α) (of A − (α)) move leftwards (rightwards).

show abstract

“…Integrals like (16) have been studied extensively in the case that Y ∈ L(H) is a Hilbert-Schmidt operator and, more generally, when Y belongs to the Schatten ideal C p (H) in L(H) for some 1 ≤ p < ∞, where they are called double operator integrals [20].…”

Section: Double Operator Integralsmentioning

confidence: 99%

“…An elementary proof of Peller's characterisation is given in Theorem 16 of Section 5 by appealing to Pisier's recent account [26] of Grothendieck's theorem. Peller's representation facilitates an explicit formula given in [20] for the trace of the integral…”

Section: Introductionmentioning

confidence: 99%

“…For a Brownian motion process b t t≥0 with respect to a probability measure P , there exists a unique X ⊗ dW in L 2 (P ⊗ P ) as X runs over all adapted simple processes; see [14], Theorem 5.3. For an adapted process X, the stochastic integral (2) is a double operator integral studied in a series of papers by Birman and Solomyak [15][16][17][18][19][20]. With the choice of the function:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Singular Bilinear Integrals in Quantum Physics

Jefferies

2015

Mathematics

View full text Add to dashboard Cite

Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how "decoupled" bilinear integration may be used to find solutions X of operator equations AX − XB = Y with respect to the spectral measure of A and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space L 1 ((P ⊗ Q) L(H) ) of double operator integrable functions for spectral measures P , Q acting in a Hilbert space H and applied to the representation of the trace of Λ×Λ ϕ d(P T P ) for a trace class operator T . The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.

show abstract

Double Operator Integrals in a Hilbert Space

Cited by 124 publications

References 14 publications

Multidimensional operator multipliers

Multidimensional operator multipliers

Untitled

Singular Bilinear Integrals in Quantum Physics

Contact Info

Product

Resources

About