SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES
|
|
- Cameron Dawson
- 5 years ago
- Views:
Transcription
1 SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SEVER S DRAGOMIR Abstract Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert spaces are given Their connection with Kittaneh s recent results which provide sharp upper and lower bounds for the numerical radius of linear operators are also established Introduction Let B H) denote the C algebra of all bounded linear operators on a complex Hilbert space H with inner product, For A B H), let w A) and A denote the numerical radius and the usual operator norm of A, respectively It is well known that w ) defines a norm on B H), and for every A B H), ) A w A) A For other results concerning the numerical range and radius of bounded linear operators on a Hilbert space, see ] and 3] In ], F Kittaneh has improved ) in the following manner: ) A A + AA w A) A A + AA, with the constants and as best possible Following Popescu s work 5], we consider the Euclidean operator radius of a pair C, D) of bounded linear operators defined on a Hilbert space H;, ) Note that in 5], the author has introduced the concept for an n tuple of operators and pointed out its main properties Let C, D) be a pair of bounded linear operators on H The Euclidean operator radius is defined by: 3) w e C, D) := sup Cx, x + Dx, x ) / x = As pointed out in 5], w e : B H) 0, ) is a norm and the following inequality holds: ) C C + D D / w e C, D) C C + D D /, where the constants and are best possible in ) Date: February, Mathematics Subject Classification Primary 7A, 7A30, 7A63 Key words and phrases Numerical radius, Euclidean radius, Bounded linear operators, Operator norm, Self-adjoint operators
2 SEVER S DRAGOMIR We observe that, if C and D are self-adjoint operators, then ) becomes 5) C + D / w e C, D) C + D / We observe also that if A B H) and A = B + ic is the Cartesian decomposition of A, then we B, C) = sup Bx, x + Cx, x ] x = By the inequality 5) and since see ]) = sup Ax, x = w A) x = 6) A A + AA = B + C ), then we have 7) 6 A A + AA w A) A A + AA We remark that the lower bound for w A) in 7) provided by Popescu s inequality ) is not as good as the first inequality of Kittaneh from ) However, the upper bounds for w A) are the same and have been proved using different arguments The main aim of this paper is to extend Kittaneh s result to Euclidean radius of two operators and investigate other particular instances of interest Related results connecting the Euclidean operator radius, the usual numerical radius of a composite operator and the operator norm are also provided Some Inequalities for the Euclidean Operator Radius The following result concerning a sharp lower bound for the Euclidean operator radius may be stated: Theorem Let B, C : H H be two bounded linear operators on the Hilbert space H;, ) Then ) w B + C )] / we B, C) B B + C C /) The constant is best possible in the sense that it cannot be replaced by a larger constant Proof We follow a similar argument to the one from ] For any x H, x =, we have ) Bx, x + Cx, x Bx, x + Cx, x ) Taking the supremum in ), we deduce B ± C) x, x 3) w e B, C) w B ± C)
3 INEQUALITIES FOR THE EUCLIDEAN RADIUS 3 Utilising the inequality 3) and the properties of the numerical radius, we have successively: we B, C) w B + C) + w B C) ] { w B + C) ] + w B C) ]} { w B + C) + B C) ]} = w B + C ), which gives the desired inequality ) The sharpness of the constant will be shown in a particular case, later on Corollary For any two self-adjoint bounded linear operators B, C on H, we have ) B + C / w e B, C) B + C /) The constant is sharp in ) Remark The inequality ) is better than the first inequality in 5) which follows from Popescu s first inequality in ) It also provides, for the case that B, C are the self-adjoint operators in the Cartesian decomposition of A, exactly the lower bound obtained by Kittaneh in ) for the numerical radius w A) Moreover, since is a sharp constant in Kittaneh s inequality ), it follows that is also the best possible constant in ) and ), respectively The following particular case may be of interest: Corollary For any bounded linear operator A : H H and α, β C we have: w α A + β A ) ] 5) α + β ) w A) α A A + β AA ) Proof If we choose in Theorem, B = αa and C = βa, we get we B, C) = α + β ) w A) and w B + C ) = w α A + β A ) ], which, by ) implies the desired result 5) Remark If we choose in 5) α = β 0, then we get the inequality 6) A + A ) w A) ) A A + AA, for any bounded linear operator A B H) If we choose in 5), α =, β = i, then we get 7) w A A ) ] w A), for every bounded linear operator A : H H
4 SEVER S DRAGOMIR The following result may be stated as well Theorem For any two bounded linear operators B, C on H we have: 8) max {w B + C), w B C)} w e B, C) w B + C) + w B C) ] / The constant is sharp in both inequalities Proof The first inequality follows from 3) For the second inequality, we observe that 9) Cx, x ± Bx, x w C ± B) for any x H, x = The inequality 9) and the parallelogram identity for complex numbers give: 0) Bx, x + Cx, x ] = Bx, x Cx, x + Bx, x + Cx, x w B + C) + w B C), for any x H, x = Taking the supremum in 9) we deduce the desired result 8) The fact that is the best possible constant follows from the fact that for B = C 0 one would obtain the same quantity w B) in all terms of 8) Corollary 3 For any two self-adjoint operators B, C on H we have: ) max { B + C, B C } w e B, C) B + C + B C ] / The constant is best possible in both inequalities Corollary Let A be a bounded linear operator on H Then { max i) A + + i) A, + i) A + i) A } ) w A) i) A + + i) A + + i) A + i) A Proof Follows from ) applied for the Cartesian decomposition of A The following result may be stated as well: ] /
5 INEQUALITIES FOR THE EUCLIDEAN RADIUS 5 Corollary 5 For any A a bounded linear operator on H and α, β C, we have: 3) max {w αa + βa ), w αa βa )} α + β ) / w A) w αa + βa ) + w αa βa ) ] / Remark 3 The above inequality 3) contains some particular cases of interest For instance, if α = β 0, then by 3) we get ) max { A + A, A A } A + A + A A ] /, w A) since, obviously w A + A ) = A + A and w A A ) = A A, A A being a normal operator Now, if we choose in 3), α = and β = i, and taking into account that A + ia and A ia are normal operators, then we get 5) max { A + ia, A ia } A + ia + A ia ] / w A) The constant is best possible in both inequalities ) and 5) The following simple result may be stated as well Proposition For any two bounded linear operators B and C on H, we have the inequality: 6) w e B, C) w C B) + w B) w C) ] / Proof For any x H, x =, we have ] Cx, x Re Cx, x Bx, x + Bx, x giving 7) = Cx, x Bx, x w C B), ] Cx, x + Bx, x w C B) + Re Cx, x Bx, x w C B) + Cx, x Bx, x for any x H, x = Taking the supremum in 7) over x =, we deduce the desired inequality 6) In particular, if B and C are self-adjoint operators, then / 8) w e B, C) B C + B C ) Now, if we apply the inequality 8) for B = A+A and C = A A i, where A B H), then we deduce:
6 6 SEVER S DRAGOMIR + i) A + i) A w A) / + A + A A A ] The following result provides a different upper bound for the Euclidean operator radius than 6) Proposition For any two bounded linear operators B and C on H, we have 9) w e B, C) min { w B), w C) } + w B C) w B + C) ] / Proof Utilising the parallelogram identity 0), we have, by taking the supremum over x H, x =, that 0) w e B, C) = w e B C, B + C) Now, if we apply Proposition for B C, B + C instead of B and C, then we can state w e B C, B + C) w C) + w B C) w B + C) giving ) w e B, C) w C) + w B C) w B + C) Now, if in ) we swap the C with B then we also have ) w e B, C) w B) + w B C) w B + C) The conclusion follows now by ) and ) A different upper bound for the Euclidean operator radius is incorporated in the following Theorem 3 Let H;, ) be a Hilbert space and B, C two bounded linear operators on H Then { 3) we B, C) max B, C } + w C B) The inequality 3) is sharp Proof Firstly, let us observe that for any y, u, v H we have successively ) y, u u + y, v v = y, u u + y, v v + Re ] y, u y, v u, v y, u u + y, v v + y, u y, v u, v y, u u + y, v v + y, u + y, v ) u, v y, u + y, v ) { max u, v } ) + u, v On the other hand, y, u + y, v ) 5) = y, u u, y + y, v v, y ] for any y, u, v H = y, y, u u + y, v v ] y y, u u + y, v v
7 INEQUALITIES FOR THE EUCLIDEAN RADIUS 7 Making use of ) and 5) we deduce that { 6) y, u + y, v y max u, v } ] + u, v for any y, u, v H, which is a vector inequality of interest in itself Now, if we apply the inequality 6) for y = x, u = Bx, v = Cx, x H, x =, then we can state that { 7) Bx, x + Cx, x max Bx, Cx } + Bx, Cx for any x H, x =, which is of interest in itself Taking the supremum over x H, x =, we deduce the desired result 3) To prove the sharpness of the inequality 3) we choose C = B, B a self-adjoint operator on H In this case, both sides of 3) become B If information about the sum and the difference of the operators B and C are available, then one may use the following result: Corollary 6 For any two operators B, C BH) we have 8) we B, C) { { max B C, B + C } } + w B C ) B + C)] The constant is best possible in 8) Proof Follows by the inequality 3) written for B + C and B C instead of B and C and by utilising the identity 0) The fact that is best possible in 8) follows by the fact that for C = B, B a self-adjoint operator, we get in both sides of the inequality 8) the quantity B Corollary 7 Let A : H H be a bounded linear operator on the Hilbert space H Then: 9) w A) { max A + A, A A } ] + w A A) A + A )] The constant is best possible Proof If B = A+A, C = A A i and Utilising 3) we deduce 9) is the Cartesian decomposition of A, then w e B, C) = w A) w C B) = w A A) A + A )] Remark If we choose in 3), B = A and C = A, A BH) then we can state that 30) w A) A + w A )] The constant is best possible in 30) Note that this inequality has been obtained in ] by the use of a different argument based on the Buzano s inequality Finally, the following upper bound for the Euclidean radius involving different composite operators also holds:
8 8 SEVER S DRAGOMIR Theorem With the assumptions of Theorem 3, we have 3) w e B, C) B B + C C + B B C C ] + w C B) The inequality 3) is sharp Proof We use 7) to write that 3) Bx, x + Cx, x Bx + Cx + Bx Cx ] + Bx, Cx for any x H, x = Since Bx = B Bx, x, Cx = C Cx, x, then 3) can be written as 33) Bx, x + Cx, x B B + C C) x, x + B B C C) x, x ] + Bx, Cx x H, x = Taking the supremum in 33) over x H, x = and noticing that the operators B B ± C C are self-adjoint, we deduce the desired result 3) The sharpness of the constant will follow from the one of 36) pointed out below Corollary 8 For any two operators B, C BH), we have 3) w e B, C) { B B + C C + B C + C B + w B C ) B + C)]} The constant is best possible Proof If we write 3) for B + C, B C instead of B, C and perform the required calculations then we get w e B + C, B C) B B + C C + B C + C B ] + w B C ) B + C)], which, by the identity 0) is clearly equivalent with 3) Now, if we choose in 3) B = C, then we get the inequality w B) B, which is a sharp inequality Corollary 9 If B, C are self-adjoint operators on H then 35) we B, C) B + C + B C ] + w CB) We observe that, if B and C are chosen to be the Cartesian decomposition for the bounded linear operator A, then we can get from 35) that 36) w A) { A A + AA + A + A ) } + w A A) A + A )] The constant is best possible This follows by the fact that for A a self-adjoint operator, we obtain in both sides of 36) the same quantity A
9 INEQUALITIES FOR THE EUCLIDEAN RADIUS 9 Now, if we choose in 3) B = A and C = A, A BH), then we get 37) w A) { A A + AA + A A AA } + w A ) This inequality is sharp The equality holds if, for instance, we assume that A is normal, ie, A A = AA In this case we get in both sides of 37) the quantity A, since for normal operators, w A ) = w A) = A Acknowledgement The author would like to thank the anonymous referee for his/her comments that have been implemented in the final version of the paper References ] SS DRAGOMIR, Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces, RGMIA Res Rep Coll, 8005), Supplement, Article 0 ONLINE ] ] KE GUSTAFSON and DKM RAO, Numerical Range, Springer-Verlag, New York, Inc, 997 3] PR HALMOS, A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, Berlin, Second edition, 98 ] F KITTANEH, Numerical radius inequalities for Hilbert space operators, Studia Math, 68) 005), ] G POPESCU, Unitary invariants in multivariable operator theory, Preprint, ArχivmathOA/009 School of Computer Science and Mathematics, Victoria University, PO Box 8, Melbourne City, VIC, 800, Australia address: severdragomir@vueduau URL:
SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES
SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. The main aim of this paper is to establish some connections that exist between the numerical
More informationINEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES
INEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. In this paper various inequalities between the operator norm its numerical radius are provided.
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More informationInequalities for the numerical radius, the norm and the maximum of the real part of bounded linear operators in Hilbert spaces
Available online at www.sciencedirect.com Linear Algebra its Applications 48 008 980 994 www.elsevier.com/locate/laa Inequalities for the numerical radius, the norm the maximum of the real part of bounded
More informationSOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR
More informationSOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES. S.S. Dragomir and M.S. Moslehian. 1. Introduction
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 23 (2008), pp. 39 47 SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES S.S. Dragomir and M.S. Moslehian Abstract. An operator T acting on
More informationarxiv: v1 [math.fa] 1 Oct 2015
SOME RESULTS ON SINGULAR VALUE INEQUALITIES OF COMPACT OPERATORS IN HILBERT SPACE arxiv:1510.00114v1 math.fa 1 Oct 2015 A. TAGHAVI, V. DARVISH, H. M. NAZARI, S. S. DRAGOMIR Abstract. We prove several singular
More informationOn the Generalized Reid Inequality and the Numerical Radii
Applied Mathematical Sciences, Vol. 5, 2011, no. 9, 441-445 On the Generalized Reid Inequality and the Numerical Radii J. O. Bonyo 1, D. O. Adicka 2, J. O. Agure 3 1,3 Department of Mathematics and Applied
More informationSOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES. 1. Introduction
SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES SEVER S. DRAGOMIR 1 AND MOHAMMAD SAL MOSLEHIAN Abstract. An oerator T is called (α, β)-normal (0 α 1 β) if α T T T T β T T. In this aer,
More informationINEQUALITIES OF LIPSCHITZ TYPE FOR POWER SERIES OF OPERATORS IN HILBERT SPACES
INEQUALITIES OF LIPSCHITZ TYPE FOR POWER SERIES OF OPERATORS IN HILBERT SPACES S.S. DRAGOMIR ; Abstract. Let (z) := P anzn be a power series with complex coe - cients convergent on the open disk D (; R)
More informationTHE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS
THE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS S.S. DRAGOMIR Abstract. Some Hermite-Hadamard s type inequalities or operator convex unctions o seladjoint operators in Hilbert spaces
More informationarxiv: v1 [math.fa] 6 Nov 2015
CARTESIAN DECOMPOSITION AND NUMERICAL RADIUS INEQUALITIES FUAD KITTANEH, MOHAMMAD SAL MOSLEHIAN AND TAKEAKI YAMAZAKI 3 arxiv:5.0094v [math.fa] 6 Nov 05 Abstract. We show that if T = H + ik is the Cartesian
More informationGeneralized Numerical Radius Inequalities for Operator Matrices
International Mathematical Forum, Vol. 6, 011, no. 48, 379-385 Generalized Numerical Radius Inequalities for Operator Matrices Wathiq Bani-Domi Department of Mathematics Yarmouk University, Irbed, Jordan
More informationMORE NUMERICAL RADIUS INEQUALITIES FOR OPERATOR MATRICES. Petra University Amman, JORDAN
International Journal of Pure and Applied Mathematics Volume 118 No. 3 018, 737-749 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.173/ijpam.v118i3.0
More informationarxiv: v1 [math.fa] 24 Oct 2018
NUMERICAL RADIUS PARALLELISM OF HILBERT SPACE OPERATORS MARZIEH MEHRAZIN 1, MARYAM AMYARI 2 and ALI ZAMANI 3 arxiv:1810.10445v1 [math.fa] 24 Oct 2018 Abstract. In this paper, we introduce a new type of
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 6 (2012), no. 1, 139 146 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/bjma/ AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH
More informationarxiv: v1 [math.fa] 1 Sep 2014
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES FOR HILBERT SPACE OPERATORS MOSTAFA SATTARI 1, MOHAMMAD SAL MOSLEHIAN 1 AND TAKEAKI YAMAZAKI arxiv:1409.031v1 [math.fa] 1 Sep 014 Abstract. We generalize
More informationDAVIS WIELANDT SHELLS OF NORMAL OPERATORS
DAVIS WIELANDT SHELLS OF NORMAL OPERATORS CHI-KWONG LI AND YIU-TUNG POON Dedicated to Professor Hans Schneider for his 80th birthday. Abstract. For a finite-dimensional operator A with spectrum σ(a), the
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: ADJOINTS IN HILBERT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: ADJOINTS IN HILBERT SPACES CHRISTOPHER HEIL 1. Adjoints in Hilbert Spaces Recall that the dot product on R n is given by x y = x T y, while the dot product on C n is
More informationEXPLICIT SOLUTION TO MODULAR OPERATOR EQUATION T XS SX T = A
EXPLICIT SOLUTION TO MODULAR OPERATOR EQUATION T XS SX T A M MOHAMMADZADEH KARIZAKI M HASSANI AND SS DRAGOMIR Abstract In this paper by using some block operator matrix techniques we find explicit solution
More informationEXPLICIT SOLUTION TO MODULAR OPERATOR EQUATION T XS SX T = A
Kragujevac Journal of Mathematics Volume 4(2) (216), Pages 28 289 EXPLICI SOLUION O MODULAR OPERAOR EQUAION XS SX A M MOHAMMADZADEH KARIZAKI 1, M HASSANI 2, AND S S DRAGOMIR 3 Abstract In this paper, by
More informationBEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction
Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of
More informationInequalities for the Numerical Radius of Linear Operators in Hilbert Spaces
This is page i Printer: Opaque this Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces Silvestru Sever Dragomir February 03 ii ABSTRACT The present monograph is focused on numerical
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationTWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES. S.S. Dragomir and J.J.
RGMIA Research Report Collection, Vol. 2, No. 1, 1999 http://sci.vu.edu.au/ rgmia TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES S.S. Dragomir and
More informationA Numerical Radius Version of the Arithmetic-Geometric Mean of Operators
Filomat 30:8 (2016), 2139 2145 DOI 102298/FIL1608139S Published by Faculty of Sciences and Mathematics, University of Niš, Serbia vailable at: htt://wwwmfniacrs/filomat Numerical Radius Version of the
More informationHERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GEOMETRICALLY CONVEX FUNCTIONS
HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GEOMETRICALLY CONVEX FUNCTIONS A TAGHAVI, V DARVISH, H M NAZARI, S S DRAGOMIR Abstract In this paper, we introduce the concept of operator geometrically
More informationNumerical Range in C*-Algebras
Journal of Mathematical Extension Vol. 6, No. 2, (2012), 91-98 Numerical Range in C*-Algebras M. T. Heydari Yasouj University Abstract. Let A be a C*-algebra with unit 1 and let S be the state space of
More informationSINGULAR VALUE INEQUALITIES FOR COMPACT OPERATORS
SINGULAR VALUE INEQUALITIES FOR OMPAT OPERATORS WASIM AUDEH AND FUAD KITTANEH Abstract. A singular value inequality due to hatia and Kittaneh says that if A and are compact operators on a complex separable
More informationA NOTE ON QUASI ISOMETRIES II. S.M. Patel Sardar Patel University, India
GLASNIK MATEMATIČKI Vol. 38(58)(2003), 111 120 A NOTE ON QUASI ISOMETRIES II S.M. Patel Sardar Patel University, India Abstract. An operator A on a complex Hilbert space H is called a quasi-isometry if
More informationarxiv: v1 [math.oa] 25 May 2009
REVERSE CAUCHY SCHWARZ INEQUALITIES FOR POSITIVE C -VALUED SESQUILINEAR FORMS MOHAMMAD SAL MOSLEHIAN AND LARS-ERIK PERSSON ariv:0905.4065v [math.oa] 5 May 009 Abstract. We prove two new reverse Cauchy
More informationINVESTIGATING THE NUMERICAL RANGE AND Q-NUMERICAL RANGE OF NON SQUARE MATRICES. Aikaterini Aretaki, John Maroulas
Opuscula Mathematica Vol. 31 No. 3 2011 http://dx.doi.org/10.7494/opmath.2011.31.3.303 INVESTIGATING THE NUMERICAL RANGE AND Q-NUMERICAL RANGE OF NON SQUARE MATRICES Aikaterini Aretaki, John Maroulas Abstract.
More informationSOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES
italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics
More informationOn multivariable Fejér inequalities
On multivariable Fejér inequalities Linda J. Patton Mathematics Department, Cal Poly, San Luis Obispo, CA 93407 Mihai Putinar Mathematics Department, University of California, Santa Barbara, 93106 September
More informationHYPO-EP OPERATORS 1. (Received 21 May 2013; after final revision 29 November 2014; accepted 7 October 2015)
Indian J. Pure Appl. Math., 47(1): 73-84, March 2016 c Indian National Science Academy DOI: 10.1007/s13226-015-0168-x HYPO-EP OPERATORS 1 Arvind B. Patel and Mahaveer P. Shekhawat Department of Mathematics,
More informationON THE GENERALIZED FUGLEDE-PUTNAM THEOREM M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI
TAMKANG JOURNAL OF MATHEMATICS Volume 39, Number 3, 239-246, Autumn 2008 0pt0pt ON THE GENERALIZED FUGLEDE-PUTNAM THEOREM M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI Abstract. In this paper, we prove
More informationSOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR PREINVEX FUNCTIONS
Banach J. Math. Anal. 9 15), no., uncorrected galleyproo DOI: 1.1535/bjma/9-- ISSN: 1735-8787 electronic) www.emis.de/journals/bjma/ SOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR
More informationA NOTE ON COMPACT OPERATORS
Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 15 (2004), 26 31. Available electronically at http: //matematika.etf.bg.ac.yu A NOTE ON COMPACT OPERATORS Adil G. Naoum, Asma I. Gittan Let H be a separable
More informationarxiv:math/ v1 [math.oa] 9 May 2005
arxiv:math/0505154v1 [math.oa] 9 May 2005 A GENERALIZATION OF ANDÔ S THEOREM AND PARROTT S EXAMPLE DAVID OPĚLA Abstract. Andô s theorem states that any pair of commuting contractions on a Hilbert space
More informationPart 1a: Inner product, Orthogonality, Vector/Matrix norm
Part 1a: Inner product, Orthogonality, Vector/Matrix norm September 19, 2018 Numerical Linear Algebra Part 1a September 19, 2018 1 / 16 1. Inner product on a linear space V over the number field F A map,
More informationis a new metric on X, for reference, see [1, 3, 6]. Since x 1+x
THE TEACHING OF MATHEMATICS 016, Vol. XIX, No., pp. 68 75 STRICT MONOTONICITY OF NONNEGATIVE STRICTLY CONCAVE FUNCTION VANISHING AT THE ORIGIN Yuanhong Zhi Abstract. In this paper we prove that every nonnegative
More informationPatryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński
Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of power-bounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF
More informationFURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM
FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM N. S. BARNETT, S. S. DRAGOMIR, AND I. S. GOMM Abstract. In this paper we establish an upper boun for the
More informationA CHARACTERIZATION OF THE OPERATOR-VALUED TRIANGLE EQUALITY
J. OPERATOR THEORY 58:2(2007), 463 468 Copyright by THETA, 2007 A CHARACTERIZATION OF THE OPERATOR-VALUED TRIANGLE EQUALITY TSUYOSHI ANDO and TOMOHIRO HAYASHI Communicated by William B. Arveson ABSTRACT.
More informationInterpolating the arithmetic geometric mean inequality and its operator version
Linear Algebra and its Applications 413 (006) 355 363 www.elsevier.com/locate/laa Interpolating the arithmetic geometric mean inequality and its operator version Rajendra Bhatia Indian Statistical Institute,
More informationInequalities for Modules and Unitary Invariant Norms
Int. J. Contemp. Math. Sciences Vol. 7 202 no. 36 77-783 Inequalities for Modules and Unitary Invariant Norms Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei
More informationResearch Article Bessel and Grüss Type Inequalities in Inner Product Modules over Banach -Algebras
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 011, Article ID 5693, 16 pages doi:10.1155/011/5693 Research Article Bessel and Grüss Type Inequalities in Inner Product Modules
More informationFIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS
Fixed Point Theory, Volume 5, No. 2, 24, 181-195 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS CEZAR AVRAMESCU 1 AND CRISTIAN
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationFunctional Analysis, Math 7321 Lecture Notes from April 04, 2017 taken by Chandi Bhandari
Functional Analysis, Math 7321 Lecture Notes from April 0, 2017 taken by Chandi Bhandari Last time:we have completed direct sum decomposition with generalized eigen space. 2. Theorem. Let X be separable
More informationSingular Value and Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017) Article 8 2017 Singular Value Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices Aliaa Burqan Zarqa University,
More informationSome Properties in Generalized n-inner Product Spaces
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 45, 2229-2234 Some Properties in Generalized n-inner Product Spaces B. Surender Reddy Department of Mathematics, PGCS, Saifabad Osmania University, Hyderabad-500004,
More informationCONVEX FUNCTIONS AND MATRICES. Silvestru Sever Dragomir
Korean J. Math. 6 (018), No. 3, pp. 349 371 https://doi.org/10.11568/kjm.018.6.3.349 INEQUALITIES FOR QUANTUM f-divergence OF CONVEX FUNCTIONS AND MATRICES Silvestru Sever Dragomir Abstract. Some inequalities
More informationCLOSED RANGE POSITIVE OPERATORS ON BANACH SPACES
Acta Math. Hungar., 142 (2) (2014), 494 501 DOI: 10.1007/s10474-013-0380-2 First published online December 11, 2013 CLOSED RANGE POSITIVE OPERATORS ON BANACH SPACES ZS. TARCSAY Department of Applied Analysis,
More informationNumerical Ranges of the Powers of an Operator
Numerical Ranges of the Powers of an Operator Man-Duen Choi 1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4 E-mail: choi@math.toronto.edu Chi-Kwong Li 2 Department
More informationDiscrete Series Representations of Unipotent p-adic Groups
Journal of Lie Theory Volume 15 (2005) 261 267 c 2005 Heldermann Verlag Discrete Series Representations of Unipotent p-adic Groups Jeffrey D. Adler and Alan Roche Communicated by S. Gindikin Abstract.
More informationSome Hermite-Hadamard type integral inequalities for operator AG-preinvex functions
Acta Univ. Sapientiae, Mathematica, 8, (16 31 33 DOI: 1.1515/ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o
More informationCORACH PORTA RECHT INEQUALITY FOR CLOSED RANGE OPERATORS
M athematical I nequalities & A pplications Volume 16, Number 2 (2013), 477 481 doi:10.7153/mia-16-34 CORACH PORTA RECHT INEQUALITY FOR CLOSED RANGE OPERATORS MARYAM KHOSRAVI Abstract. By B(H ) we denote
More informationAPPROXIMATE WEAK AMENABILITY OF ABSTRACT SEGAL ALGEBRAS
MATH. SCAND. 106 (2010), 243 249 APPROXIMATE WEAK AMENABILITY OF ABSTRACT SEGAL ALGEBRAS H. SAMEA Abstract In this paper the approximate weak amenability of abstract Segal algebras is investigated. Applications
More informationWhere is matrix multiplication locally open?
Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?
More informationON k QUASI CLASS Q OPERATORS (COMMUNICATED BY T. YAMAZAKI)
Bulletin of Mathematical Analysis and Applications ISSN: 181-191, URL: http://www.bmathaa.org Volume 6 Issue 3 (014), Pages 31-37. ON k QUASI CLASS Q OPERATORS (COMMUNICATED BY T. YAMAZAKI) VALDETE REXHËBEQAJ
More informationEXPLICIT SOLUTION OF THE OPERATOR EQUATION A X + X A = B
EXPLICIT SOLUTION OF THE OPERATOR EQUATION A X + X A = B Dragan S. Djordjević November 15, 2005 Abstract In this paper we find the explicit solution of the equation A X + X A = B for linear bounded operators
More informationMaximizing the numerical radii of matrices by permuting their entries
Maximizing the numerical radii of matrices by permuting their entries Wai-Shun Cheung and Chi-Kwong Li Dedicated to Professor Pei Yuan Wu. Abstract Let A be an n n complex matrix such that every row and
More informationj=1 [We will show that the triangle inequality holds for each p-norm in Chapter 3 Section 6.] The 1-norm is A F = tr(a H A).
Math 344 Lecture #19 3.5 Normed Linear Spaces Definition 3.5.1. A seminorm on a vector space V over F is a map : V R that for all x, y V and for all α F satisfies (i) x 0 (positivity), (ii) αx = α x (scale
More informationAbstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization relations.
HIROSHIMA S THEOREM AND MATRIX NORM INEQUALITIES MINGHUA LIN AND HENRY WOLKOWICZ Abstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationFUNCTIONAL ANALYSIS-NORMED SPACE
MAT641- MSC Mathematics, MNIT Jaipur FUNCTIONAL ANALYSIS-NORMED SPACE DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR 1. Normed space Norm generalizes the concept of length in an arbitrary
More informationHigher rank numerical ranges of rectangular matrix polynomials
Journal of Linear and Topological Algebra Vol. 03, No. 03, 2014, 173-184 Higher rank numerical ranges of rectangular matrix polynomials Gh. Aghamollaei a, M. Zahraei b a Department of Mathematics, Shahid
More informationRemarks on Fuglede-Putnam Theorem for Normal Operators Modulo the Hilbert-Schmidt Class
International Mathematical Forum, Vol. 9, 2014, no. 29, 1389-1396 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47141 Remarks on Fuglede-Putnam Theorem for Normal Operators Modulo the
More information1 Fourier Transformation.
Fourier Transformation. Before stating the inversion theorem for the Fourier transformation on L 2 (R ) recall that this is the space of Lebesgue measurable functions whose absolute value is square integrable.
More informationMeans of unitaries, conjugations, and the Friedrichs operator
J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More informationarxiv: v1 [math.fa] 26 May 2012
EXPONENTIALS OF BOUNDED NORMAL OPERATORS arxiv:1205.5888v1 [math.fa] 26 May 2012 AICHA CHABAN 1 AND MOHAMMED HICHEM MORTAD 2 * Abstract. The present paper is mainly concerned with equations involving exponentials
More informationClarkson Inequalities With Several Operators
isid/ms/2003/23 August 14, 2003 http://www.isid.ac.in/ statmath/eprints Clarkson Inequalities With Several Operators Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute, Delhi Centre 7, SJSS Marg,
More informationNumerical Ranges and Dilations. Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement
Numerical Ranges and Dilations Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement Man-Duen Choi Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3.
More informationLecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why?
KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices Why? Lecture 5 Ch. 5, Norms for vectors and matrices Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Problem: Measure size of
More informationarxiv: v1 [math.fa] 19 Aug 2017
EXTENSIONS OF INTERPOLATION BETWEEN THE ARITHMETIC-GEOMETRIC MEAN INEQUALITY FOR MATRICES M. BAKHERAD 1, R. LASHKARIPOUR AND M. HAJMOHAMADI 3 arxiv:1708.0586v1 [math.fa] 19 Aug 017 Abstract. In this paper,
More informationPRODUCT OF OPERATORS AND NUMERICAL RANGE
PRODUCT OF OPERATORS AND NUMERICAL RANGE MAO-TING CHIEN 1, HWA-LONG GAU 2, CHI-KWONG LI 3, MING-CHENG TSAI 4, KUO-ZHONG WANG 5 Abstract. We show that a bounded linear operator A B(H) is a multiple of a
More informationON SOME INEQUALITIES IN NORMED LINEAR SPACES
ON SOME INEQUALITIES IN NORMED LINEAR SPACES S.S. DRAGOMIR Abstract. Upper ad lower bouds for the orm of a liear combiatio of vectors are give. Applicatios i obtaiig various iequalities for the quatities
More informationWavelets and Linear Algebra
Wavelets and Linear Algebra 4(1) (2017) 43-51 Wavelets and Linear Algebra Vali-e-Asr University of Rafsanan http://walavruacir Some results on the block numerical range Mostafa Zangiabadi a,, Hamid Reza
More informationDIAGONAL TOEPLITZ OPERATORS ON WEIGHTED BERGMAN SPACES
DIAGONAL TOEPLITZ OPERATORS ON WEIGHTED BERGMAN SPACES TRIEU LE Abstract. In this paper we discuss some algebraic properties of diagonal Toeplitz operators on weighted Bergman spaces of the unit ball in
More informationFURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL FOR POWER SERIES IN BANACH ALGEBRAS VIA GRÜSS-LUPAŞ TYPE INEQUALITIES FOR p-norms
Mem. Gra. Sci. Eng. Shimane Univ. Series B: Mathematics 49 06), pp. 5 34 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL FOR POWER SERIES IN BANACH ALGEBRAS VIA GRÜSS-LUPAŞ TYPE INEQUALITIES FOR p-norms SILVESTRU
More informationON THE UNILATERAL SHIFT AND COMMUTING CONTRACTIONS
ON THE UNILATERAL SHIFT AND COMMUTING CONTRACTIONS JUSTUS K. MILE Kiriri Women s University of Science & Technology, P. O. Box 49274-00100 Nairobi, Kenya Abstract In this paper, we discuss the necessary
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationIntertibility and spectrum of the multiplication operator on the space of square-summable sequences
Intertibility and spectrum of the multiplication operator on the space of square-summable sequences Objectives Establish an invertibility criterion and calculate the spectrum of the multiplication operator
More informationSome results on the reverse order law in rings with involution
Some results on the reverse order law in rings with involution Dijana Mosić and Dragan S. Djordjević Abstract We investigate some necessary and sufficient conditions for the hybrid reverse order law (ab)
More informationAN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX
Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX TIN-YAU TAM AND HUAJUN HUANG Abstract.
More informationON A CHARACTERISATION OF INNER PRODUCT SPACES
Georgian Mathematical Journal Volume 8 (2001), Number 2, 231 236 ON A CHARACTERISATION OF INNER PRODUCT SPACES G. CHELIDZE Abstract. It is well known that for the Hilbert space H the minimum value of the
More informationHilbert Spaces. Contents
Hilbert Spaces Contents 1 Introducing Hilbert Spaces 1 1.1 Basic definitions........................... 1 1.2 Results about norms and inner products.............. 3 1.3 Banach and Hilbert spaces......................
More informationYimin Wei a,b,,1, Xiezhang Li c,2, Fanbin Bu d, Fuzhen Zhang e. Abstract
Linear Algebra and its Applications 49 (006) 765 77 wwwelseviercom/locate/laa Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices Application of perturbation theory
More informationMatrix Inequalities by Means of Block Matrices 1
Mathematical Inequalities & Applications, Vol. 4, No. 4, 200, pp. 48-490. Matrix Inequalities by Means of Block Matrices Fuzhen Zhang 2 Department of Math, Science and Technology Nova Southeastern University,
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationThe singular value of A + B and αa + βb
An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) Tomul LXII, 2016, f. 2, vol. 3 The singular value of A + B and αa + βb Bogdan D. Djordjević Received: 16.II.2015 / Revised: 3.IV.2015 / Accepted: 9.IV.2015
More informationarxiv: v2 [math.cv] 11 Mar 2016 BEYAZ BAŞAK KOCA AND NAZIM SADIK
INVARIANT SUBSPACES GENERATED BY A SINGLE FUNCTION IN THE POLYDISC arxiv:1603.01988v2 [math.cv] 11 Mar 2016 BEYAZ BAŞAK KOCA AND NAZIM SADIK Abstract. In this study, we partially answer the question left
More informationCharacterization of half-radial matrices
Characterization of half-radial matrices Iveta Hnětynková, Petr Tichý Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8, Czech Republic Abstract Numerical radius r(a) is the
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More informationarxiv: v1 [math.fa] 30 Oct 2011
AROUND OPERATOR MONOTONE FUNCTIONS MOHAMMAD SAL MOSLEHIAN AND HAMED NAJAFI arxiv:111.6594v1 [math.fa] 3 Oct 11 Abstract. We show that the symmetrized product AB + BA of two positive operators A and B is
More informationSection 3.9. Matrix Norm
3.9. Matrix Norm 1 Section 3.9. Matrix Norm Note. We define several matrix norms, some similar to vector norms and some reflecting how multiplication by a matrix affects the norm of a vector. We use matrix
More information3 Orthogonality and Fourier series
3 Orthogonality and Fourier series We now turn to the concept of orthogonality which is a key concept in inner product spaces and Hilbert spaces. We start with some basic definitions. Definition 3.1. Let
More informationPROJECTIONS ONTO CONES IN BANACH SPACES
Fixed Point Theory, 19(2018), No. 1,...-... DOI: http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html PROJECTIONS ONTO CONES IN BANACH SPACES A. DOMOKOS AND M.M. MARSH Department of Mathematics and Statistics
More information