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.

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 Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

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 On the Relation between States and Maps in Infinite Dimensions

2007 ◽  
Vol 14 (04) ◽  
pp. 355-370 ◽  
Author(s):  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
Hilbert Spaces ◽  
Trace Class ◽  
Quantum Systems ◽  
Bounded Operators ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Continuous Map ◽  
Finite Dimensional ◽  
Trace Class Operators ◽  
Product Realization

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

scholarly journals Schur Algebras overC*-Algebras

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong ◽  
Sheng-Wang Wang
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Trace Class ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Schur Algebras ◽  
Product Operation ◽  
The Matrix ◽  
Trace Class Operators

Let&#x1D49C;be aC*-algebra with identity1, and lets(&#x1D49C;)denote the set of all states on&#x1D49C;. Forp,q,r∈[1,∞), denote by&#x1D4AE;r(&#x1D49C;)the set of all infinite matricesA=[ajk]j,k=1∞over&#x1D49C;such that the matrix(ϕ[A[2]])[r]:=[(ϕ(ajk*ajk))r]j,k=1∞defines a bounded linear operator fromℓptoℓqfor allϕ∈s(&#x1D49C;). Then&#x1D4AE;r(&#x1D49C;)is a Banach algebra with the Schur product operation and norm‖A‖=sup{‖(ϕ[A[2]])r‖1/(2r):ϕ∈s(&#x1D49C;)}. Analogs of Schatten's theorems on dualities among the compact operators, the trace-class operators, and all the bounded operators on a Hilbert space are proved.

scholarly journals Range-Kernel Orthogonality and Elementary Operators: The Nonsmoothness Case

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
A. Bachir ◽  
A. Segres ◽  
Nawal Sayyaf
Keyword(s):  
Trace Class ◽  
Smooth Case ◽  
Counter Example ◽  
Elementary Operators ◽  
Trace Class Operators

The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the range of elementary operators has been carried out for certain kinds of elementary operators by many authors in the smooth case. In this note, we study that the characterization is a problem in nonsmoothness case for general elementary operators, and we give a counter example to S. Mecheri and M. Bounkhel results.

scholarly journals Fisher information for inverse problems and trace class operators

2012 ◽  
Vol 53 (12) ◽  
pp. 123503 ◽  
Author(s):  
S. Nordebo ◽  
M. Gustafsson ◽  
A. Khrennikov ◽  
B. Nilsson ◽  
J. Toft
Keyword(s):  
Inverse Problems ◽  
Fisher Information ◽  
Trace Class ◽  
Trace Class Operators

Trace Class Elements and Cross-Sections in Kac-Moody Groups

1998 ◽  
Vol 50 (5) ◽  
pp. 972-1006 ◽  
Author(s):  
Gerd Brüchert
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Trace Class ◽  
Irreducible Representations ◽  
Functional Identity ◽  
Trace Class Operators ◽  
Regular Classes

AbstractLet G be an affine Kac-Moody group, π0, … ,πr, πδ its fundamental irreducible representations and χ0, … , χr, χδ their characters. We determine the set of all group elements x such that all πi(x) act as trace class operators, i.e., such that χi(x) exists, then prove that the χ i are class functions. Thus, χ := (χ0, … , χr, χδ) factors to an adjoint quotient χ for G. In a second part, following Steinberg, we define a cross-section C for the potential regular classes in G. We prove that the restriction χ|C behaves well algebraically. Moreover, we obtain an action of C ℂ✗ on C, which leads to a functional identity for χ|C which shows that χ|C is quasi-homogeneous.

scholarly journals Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc

2020 ◽  
pp. 1-18
Author(s):  
Augustin Mouze ◽  
Vincent Munnier
Keyword(s):  
Critical Exponent ◽  
Analytic Functions ◽  
Unit Disc ◽  
Optimal Growth ◽  
Space Of Analytic Functions ◽  
Shift Operators ◽  
Image Position

Abstract For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$ $ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$ We establish the optimal growth of frequently hypercyclic functions for $T_\alpha $ in terms of $L^p$ averages, $1\leq p\leq +\infty $ . This allows us to highlight a critical exponent.

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