Grüss-Landau inequalities for elementary operators and inner product type transformers in Q and Q* norm ideals of compact operators

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2447-2455
Author(s):  
Milan Lazarevic
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Compact Operators ◽  
Product Type ◽  
Inner Product ◽  
Operator Inequality ◽  
Space Operator ◽  
Bounded Operators ◽  
Normal Operators ◽  
Elementary Operators

For a probability measure ? on ? and square integrable (Hilbert space) operator valued functions {A*t}t??, {Bt}t??, we prove Gr?ss-Landau type operator inequality for inner product type transformers |?? AtXBtd?(t)- ?? At d?(t)X ?? Btd?(t)|2? ? ||??AtA*td?(t)- |??A*td?(t)|2||? (?? B*tX*XBtd?(t)- |X?? Btd?(t)|2)?, for all X ? B(H) and for all ? ? [0,1]. Let p ? 2, ? to be a symmetrically norming (s.n.) function, ? (p) to be its p-modification, ? (p)* is a s.n. function adjoint to ?(p) and ||?||?(p)* to be a norm on its associated ideal C?(p)*(H) of compact operators. If X ? C?(p)*(H) and {?n}?n=1 is a sequence in (0,1], such that ??n=1 ?n = 1 and ??n=1 ||?n-1/2 An f||2+||?-1/2n B*nf||2 < +? for some families {An}? n=1 and {Bn}? n=1 of bounded operators on Hilbert space H and for all f ? H, then ||?? n=1 ?-1n AnXBn-?? n=1 AnX ?? n=1 Bn||?(p)* ? ||???n=1 ?-1n |An|2-|??n=1 An|2 X ? ??n=1 ?-1n |B*n|2-|??n=1 B*n|2||?(p)+, if at least one of those operator families consists of mutually commuting normal operators. The related Gr?ss-Landau type ||?||?(p) norm inequalities for inner product type transformers are also provided.

scholarly journals Norm inequalities for elementary operators and other inner product type integral transformers with the spectra contained in the unit disc

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Danko Jocic ◽  
Stefan Milosevic ◽  
Vladimir Djuric
Keyword(s):  
Unit Disc ◽  
Trace Class ◽  
Inner Product ◽  
Bounded Operators ◽  
Shift Operators ◽  
Hilbert Space Operators ◽  
Type Integral ◽  
Normal Operators ◽  
Elementary Operators ◽  
Trace Class Operators

If {At}t?? and {Bt}t?? are weakly*-measurable families of bounded Hilbert space operators such that transformers X ??? At*XAtd?(t) and X ??? Bt*XBtd?(t) on B(H) have their spectra contained in the unit disc, then for all bounded operators X ||?AX?B|| ? ||X-?? At* XBtd?(t)||, (1) where ?A def= s-limr?1(I + ??,n=1 r2n ??...??|At1...Atn|2d?n(t1,...,tn)-1/2 and ?B by analogy. If additionally ??,n=1 ??n |A*t1...A*tn|2d?n(t1,...,tn) and ?,n=1 ??n|B*t1...B*tn|2d?n(t1,...,tn) both represent bounded operators, then for all p,q,s > 1 such that 1/q + 1/s = 2/p and for all Schatten p trace class operators X ||?1-1q,A X ?1-1s,B|| ? ||?-1/q,A*(X-?? At* XBtd?(t))?-1/s,B*||p.(2) If at least one of those families consists of bounded commuting normal operators, then (1) holds for all unitarily invariant Q-norms. Applications to the shift operators are also given.

scholarly journals Schatten's theorems on functionally defined Schur algebras

2005 ◽  
Vol 2005 (14) ◽  
pp. 2175-2193 ◽  
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Bounded Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Bounded Operators ◽  
Schur Algebras ◽  
Schur Product ◽  
Product Operation ◽  
Trace Class Operators

For each triple of positive numbersp,q,r≥1and each commutativeC*-algebraℬwith identity1and the sets(ℬ)of states onℬ, the set&#x1D4AE;r(ℬ)of all matricesA=[ajk]overℬsuch thatϕ[A[r]]:=[ϕ(|ajk|r)]defines a bounded operator fromℓptoℓqfor allϕ∈s(ℬ)is shown to be a Banach algebra under the Schur product operation, and the norm‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the&#x1D4AE;r(ℬ)setting.

scholarly journals Antinormal composition operators on $ \mbf{\ell^2}$

2008 ◽  
Vol 39 (4) ◽  
pp. 347-352 ◽  
Author(s):  
Gyan Prakash Tripathi ◽  
Nand Lal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Composition Operators ◽  
Complete Characterization ◽  
Inner Product ◽  
Complex Numbers ◽  
Bounded Linear ◽  
Normal Operators

A bounded linear operator $ T $ on a Hilbert space $ H $ is called antinormal if the distance of $ T $ from the set of all normal operators is equal to norm of $ T $. In this paper, we give a complete characterization of antinormal composition operators on $ \ell^2 $, where $ \ell^2 $ is the Hilbert space of all square summable sequences of complex numbers under standard inner product on it.

scholarly journals Completely positive maps for imprimitive complex reflection groups

2021 ◽  
Vol 13 (2) ◽  
pp. 452-459
Author(s):  
H. Randriamaro
Keyword(s):  
Hilbert Space ◽  
Coxeter Groups ◽  
Fock Space ◽  
Inner Product ◽  
Reflection Groups ◽  
Bounded Operators ◽  
Completely Positive ◽  
Creation And Annihilation Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.

Strong Extensions vs. Weak Extensions of C*-Algebras

1978 ◽  
Vol 21 (2) ◽  
pp. 143-147
Author(s):  
S. J. Cho
Keyword(s):  
Hilbert Space ◽  
Partial Isometry ◽  
Compact Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Image Position ◽  
The Ideal

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such thatfor all a in A.

On a Characterisation of Inner Product Spaces

2001 ◽  
Vol 8 (2) ◽  
pp. 231-236
Author(s):  
G. Chelidze
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Normed Space ◽  
Inner Product ◽  
Product Spaces ◽  
Second Moment ◽  
Inner Product Spaces ◽  
Minimum Value ◽  
Positive Weights ◽  
The Mean

Abstract It is well known that for the Hilbert space H the minimum value of the functional F μ (f) = ∫ H ‖f – g‖2 dμ(g), f ∈ H, is achived at the mean of μ for any probability measure μ with strong second moment on H. We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.

On Decomposability of Compact Perturbations of Normal Operators

1975 ◽  
Vol 27 (3) ◽  
pp. 725-735 ◽  
Author(s):  
M. Radjabalipour ◽  
H. Radjavi
Keyword(s):  
Hilbert Space ◽  
Characteristic Function ◽  
Imaginary Part ◽  
Contraction Operator ◽  
Cayley Transform ◽  
Space Operator ◽  
Schatten Class ◽  
Compact Perturbations ◽  
Normal Operators ◽  
Decomposable Operator

The main purpose of this paper is to show that a bounded Hilbert-space operator whose imaginary part is in the Schatten class Cp(1 ≦ p < ∞ ) is strongly decomposable. This answers affirmatively a question raised by Colojoara and Foias [6, Section 5(e), p. 218].In case 0 ≦ T* — T ∈ C1, it was shown by B. Sz.-Nagy and C. Foias [2, p. 442; 25, p. 337] that T has many properties analogous to those of a decomposable operator and by A. Jafarian [11] that T is strongly decomposable. The authors of [11] and [24] employ the properties of the characteristic function of the contraction operator obtained from the Cayley transform of T;

scholarly journals Elementary operators on prime C*-algebras II

1988 ◽  
Vol 30 (3) ◽  
pp. 275-284 ◽  
Author(s):  
Martin Mathieu
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Regular Representation ◽  
Bounded Operators ◽  
Weakly Compact ◽  
C Algebras ◽  
Left Regular Representation ◽  
Left Regular ◽  
Elementary Operators ◽  
The Right

Compact elementary operators acting on the algebra ℒ(H) of all bounded operators on some Hilbert space H were characterised by Fong and Sourour in [9]. Akemann and Wright investigated compact and weakly compact derivations on C*-algebras [1], and also studied compactness properties of the sum and the product of the right and the left regular representation of an element in a C*-algebra [2]. They used the concept of a compact Banach algebra element due to Vala [17]: an element a in a Banach algebra A is called compact if the mapping x → axa is compact on A. This notion has been further investigated by Ylinen [18, 19, 20], who showed in particular that a is a compact element of the C*-algebra A if x ↦ axa is weakly compact on A [19].

On Permutability and Submultiplicativity of Spectral Radius

1995 ◽  
Vol 47 (5) ◽  
pp. 1007-1022 ◽  
Author(s):  
W. E. Longstaff ◽  
H. Radjavi
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Semigroup Of Operators ◽  
Multiplicative Semigroup ◽  
Orthogonal Projections ◽  
Normal Operators

AbstractLet r(T) denote the spectral radius of the operator T acting on a complex Hilbert space H. Let S be a multiplicative semigroup of operators on H. We say that r is permutable on 𝓢 if r(ABC) = r(BAC), for every A,B,C ∈ 𝓢. We say that r is submultiplicative on 𝓢 if r(AB) ≤ r(A)r(B), for every A, B ∈ 𝓢. It is known that, if r is permutable on 𝓢, then it is submultiplicative. We show that the converse holds in each of the following cases: (i) 𝓢 consists of compact operators (ii) 𝓢 consists of normal operators (iii) 𝓢 is generated by orthogonal projections.

scholarly journals On the class of (A,n) - real power positive operators in semi-hilbertian space

2019 ◽  
Vol 25 (2) ◽  
pp. 161-166
Author(s):  
Abdelkader Benali
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Positive Operator ◽  
Primary 47B20 ◽  
Positive Operators ◽  
Subject Classification ◽  
Inner Product ◽  
Space Operator ◽  
Real Power ◽  
Oblique Projections

In this paper, the concept of the class of n-Real power positive operators on a hilbert space defined by Abdelkader Benali in [1] is generalized when an additional semi-inner product is considered. This new concept is described by means of oblique projections. For a Hilbert space operator T ∈ B(H) is (A,n) - Real power positive operators for some positive operator A and for some positive integer n ifTn + T#n ≥A 0, n = 1,2,...Keywords: Real power, Semi-Hilbertian space, Semi-inner product, Positive operators. 2000Mathematics Subject Classification: Primary 47B20. Secondary 47B99

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