DOCSLIB.ORG

Integral Equations and Operator Theory Vol. 12 (1989) 0378-620X

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Integral Equations and Operator Theory Vol. 12 (1989) 0378-620X Integral Equations 0378-620X/89/020155-0851.50+0.20/0 and Operator Theory (e) 1989 Birkh~user Verlag, Basel Vol. 12 (1989) A GENERALIZATION OF A THEOREM OF AMEMIYA AND ANDO ON THE CONVERGENCE OF RANDOM PRODUCTS OF CONTRACTIONS IN HILBERT SPACE John Dye Let {T 1, ..., TN} be a finite set of linear contraction mappings of a Hilbert space Hinto itself, and let r~oe a mapping from the natural numbers N to { 1, ..., N} which assumes each value infinitely often. One can form S n = Tr(n) "'" Tr(1) which could be described as a random product of the Ti's. If the contractions have t'h6 condifibn (W): IlWxll < I~1 whenever Tx ~ x, then S n converges weakly to the projection Q onto the subspace ni_~l~ [ x I Tix ='x]. This theorem is due to Amemiya and Ando. We demonstrate a basic prbperty of the algebraic semigroup S = S(T 1 ..... TN) generated by N contractions, each having (W). We prove that if the semigroup of an infinite set of contractions is equipped with this property, and the maps satisfy a minor condition parallel to (W) on each of N maps, then random products still converge weakly. Our proof is different from Amemiya and Ando's. We illustrate our method with a new proof of the fact that if a contraction T is completely non-unitary, then T n ---) 0 weakly. 1. INTRODUCTION. If P and Q are orthogonal projections onto subspaces Sp and SQ, respectively, in a Hilbert space H, then the sequence of products P, QP, PQP, QPQP, ... converges strongly to the projection onto the subspace Sp n SQ. This was known to J. yon Neumann in 1933; see [6], More generally, letting {P1, P2 ..... PN} be a finite set of projections, and r a mapping from N to {1,2 ..... N}, we can form S n = Pr(n) "'" Pr(2)Pr(1) This is called a random product of the Pi's. Independently, F. Browder (giving credit to S. Kakutani) ([ 2], 1958) and M. Prager ([7], 1960) proved the following: suppose that the selection r is quasi-periodic, in the sense that, for some fixed k > 0, each block of k consecutive values of r contains all values 1, 2, ..., N. Then S n converges weakly to the orthogonal projection Q onto the subspace of vectors fixed under each Pi" In 1965, Amemiya and Ando [1] proved that the periodicity assumptions could be 156 Dye dropped, and moreover, that only the condition (W) : IlTxll < Ilxll whenever Tx # x, on each of N contractions, is sufficient for weak convergence of the random product. Evidently, condition (W) appears to have been first used by Halperin ([4], 1962). It is of interest to note that results in the Amemiya and Ando paper have been applied to computer tomography. (See [9].) Amemiya and Ando's proof involved induction (on the finite number of maps.) Although ingenious, it was complicated by the fact that the "new map" can occur anywhere in a given sequence, an infinite number of times! It is natural to ask if a different style of proof, necessarily not based on induction, might accommodate random products from an infinite set of contractions. Our approach reveals a striking uniform condition on the algebraic semigroup S generated by a finite number of (W) contractions, which we shall call (USC): for any sequence of bounded vectors Vn---) 0 weakly, and any sequences of letters (Tn) and words (Wn) from S, Tn*Ynvn~ 0 weakly, and, if [[Vnl[ - [IWn Vnll --~ O, then W n v n ~ 0 weakly, also. Trivial counterexamples (to weak convergence) exist when the maps in the random product come from an infinite set of projections. However, in the presence of (USC), we show that a modest variation of the assumption of (W) for each of N maps to an infinite set of maps gives weak convergence of the product. ff a contraction T satisfies ( [[Tnx[[ = IlT*nx[[ = Ilx[[for all n > 0 imples x = 0) it is called completely non-unitary. (See Sz.-Nagy and Foias [5].) We will designate this class by (CN). In the next section, we demonstrate our technique for the main theorem by showing that a product of a single (CN) map converges weakly to zero. This was probably first proved by Foguel [3]. A different proof appears in Amemiya and Ando [1]. 2. THE WEAK CONVERGENCE OF A SINGLE (CN) CONTRACTION. LEMMA 1. Let (Xn) be a bounded sequence of veclors in a Hilbert space which does not converge weakly to zero. By dropping to an appropriaIe subsequence, we may assume that Xn = ff-n u + v n, where u ~ 0, t/n---> 1, u _k v n V n, and v n ---> 0 weakly. Proof. Since (Xn) does not converge weakly to zero, there exists some w ~ H, some a > 0, and for some subsequence we may assume (2.1) [(xn, w)l > a. Let SUPn ( IlXnll ) = b < oo. By the weak compactness of B(0; b) we can drop to a convergent further subsequence of (Xn) and assume (2.2) x n ~ u weakly (which by (2.1) is not zero). Dye 157 Since H = < u > @ < u > _1_, we can decompose x n = e.nu + Vn, where v n < u >2_. To prove the Lemma, it suffices to show that CZn---~ 1 and v n --~ 0 weakly. Let 0 ~ z H be arbitrary, and decompose z as above by writing z = c~u + 13 v, where v < u > _k. (2.2) implies (x n - u, z) = (~nu + v n - u, ocu + 13v) = (2.3) ((~a- 1 ) u + v n, czu + 13v ) ---> 0. If 13 = 0 then (2.4) (Vn, cz u) = 0 and (2.3) gives cz(ct,n - 1) Ilull2 ~ 0, proving that or.n ---) 1. Ifo~ = 0 then (2.3) gives ((%-l)u+v n,13v) = (2.5) (v n, 13v) ---> 0. Hence (Vn, z)= (v n,czu+13v) = (vla, czu) + (v n,13v) = 0 + (v n,13v) --) 0 by (2.4) and (2.5). Thus v n ~ 0 weakly. PROPOSITION 1. (Foguel [3]). Let T have condition (CN). Then T n converges weakly to zero. Proof. We assume for some x that T n x does not converge weakly to zero, and arrive at a contradiction. By Lemma 1 we can drop to a subsequence and assume Tnx = crn u + v n, where u # 0, r n ~ 1, v n ~ 0 weakly and u .1_ v n. Note that IlTnxl[, lITn-k xf[, and liTn+k xl[ must all share the same nonzero limit L = ( I[ull2 + lim n [[Vnl]2)1/2. Let k _> 0 be fixed. We have (2.6) (Tnx, Tnx) = (Tn-kx, T *k (o~n u + Vn) ) and (2.7) (Wn+k x, TkTnx) = (Tn+kx, T k (czn u + Vn)). Applying the Schwarz inequality to the right hand side of (2.6) and (2.7), and taking limits as n ~ ~, we get by v n ----r0 weakly (2.6') L < ( lIT*k ul[ 2 + lim SUPn [[T*kvn[[ 2 )1/2 and (2.7') L _ ( lit k ull 2 + lim SUPnl[Tk v n II2 )112 Since both expressions on the right hand sides of (2.6') and (2.7') are < L, we must have that itT*kutl = I/utl and ilTku[I = Ilufl. Since k was arbitrary and T has condition (CN) we conclude that u must be zero, contradicting our assumption that Tnx does not converge weakly to zero. 3. RANDOM PRODUCTS OF CONTRACTIONS. PROPOSITION 2. Let S = S(T 1 ..... TN) be the algebraic semigroup generated by N linear contractions in a Hilbert space, each having (W). Then S has (USC). Proof. We note first that since T n can take on at most N distinct maps, the condition Tn*T n Vn--> 0 is satisfied for (bounded) vectors v n ---> 0. Let Q be the projection onto the subspace NQ = ~ kl'I=l F(Tk). Then we can decompose the v n as Vn(1) + Vn(2), where 158 Dye Vn(1) a QH and Vn(2) e (I - Q)H. Expanding (3.1) Ilvnll2- IlWnvnll2, decomposing v n, and recalling that a vector fixed by a contraction is also fixed by its adjoint (see [7]), we see that (3.1) ~ 0 implies that Ilvn(2)l[2 - I[Wnvn(2)[I 2 ~ 0. Hence if (USC) holds on (I - Q)H, then the condition IlvnlI - ]lWnvn[ I ~ 0 will imply Wnv n ~ 0 weakly on all of H, as (USC) holds automatically on QH. To complete the proof, we will replace (I - Q)H by H and assume NQ -- (0). We proceed by induction, considering the case N = 1 first. Suppose that Tk(n) v n does not converge weakly to zero, for some bounded v n --> 0 weakly and some map k(n): N to N, even though Ilvnll - IITK(n) vnll ~ 0. We can drop to a subsequence and assume k(n) > 1. Necessarily Tk(n) -1 v n does not converge weakly to zero, so we can employ Lemma 1, and for a subsequence assume (3.2) Tk(n)-I Vn = anu + qn where u ~ 0, oqa---> 1, u .1_ qn Vn, and qn---> 0 weakly. We now compute (3.3) IITk(n)-I v n II2 - Ilwk(n)vnll 2 = (3.4) I[O~nUlL2- IIU.nTull2 + (3.5) [Iqnll2- IlTqnl[2 + (3.6) (qn, ~ u) + (0tn u, qn) (Tqn, a n Tu) (o~n Tu, Tqn ). As qn --* 0 weakly, (3.6) ~ 0. Because [[vnll - IlTk(n) Vn[I ~ 0, we note that (3.3) ~ 0.
Load more

Recommended publications
  • Operators Similar to Their Adjoints
    ––––––––––––––––––––––––––––— Popular and Democratic Republic of Algeria University of Oran 1 Ahmed Ben Bella Faculty of exact and applied Sciences Department of Mathematics Operators Similar To Their Adjoints THESIS SUBMITTED FOR THE DEGREE OF DOCTORATE IN MATHEMATICS Presented by : Dehimi Souheyb Doctorate’sThesis Committee : President: Professor C.Bouzar University of Oran 1 Ahmed Ben Bella Thesis Supervisor: Professor M.H.Mortad University of Oran 1 Ahmed Ben Bella Examiners: Professor K.Belghaba University of Oran 1 Ahmed Ben Bella Dr. M.Meftah (M.C.A) University of Oran 1 Ahmed Ben Bella Professor B.Benahmed National Polytechnic School of Oran (ENPO Ex: ENSET) Professor M.Tlemçani University of Sciences and Technology of Oran (M.B) Academic year 2016 2017 Acknowledgement I would like to thank first and for most my supervisor, Professor Mohammed Hichem Mortad, for his many suggestions, helpful discussions, patience and constant support during this research. I sincerely thank Professor C.Bouzar for giving me the honor of being president of the jury. I thank Professors K.Belghaba, B.Benahmed and M.Tlemçani and Doctor M.Meftah for their time and effort participating in my thesis committee. Of course, I am grateful to my parents for their support, encouragement, and patience. Without them this work would never have come into existence. Contents Introduction vi 1 Essential background 11 1.1 Banachalgebra ................................. 11 1.1.1 Introduction............................... 11 1.1.2 Basic properties of spectra . 12 1.2 C-algebra.................................... 13 1.3 Boundedoperators ............................... 15 1.3.1 Definitions and properties . 17 1.3.2 Approximate point spectrum . 18 1.3.3 Resolutions of the identity .
    [Show full text]
  • Unbounded Quasinormal Operators Revisited
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE Integr. Equ. Oper. Theory 79 (2014), 135–149 provided by Springer - Publisher Connector DOI 10.1007/s00020-014-2133-1 Published online March 6, 2014 c The Author(s) This article is published Integral Equations with open access at Springerlink.com 2014 and Operator Theory Unbounded Quasinormal Operators Revisited Zenon Jan Jablo´nski, Il Bong Jung and Jan Stochel Abstract. Various characterizations of unbounded closed densely defined operators commuting with the spectral measures of their moduli are established. In particular, Kaufman’s definition of an unbounded quasi- normal operator is shown to coincide with that given by the third-named author and Szafraniec. Examples demonstrating the sharpness of results are constructed. Mathematics Subject Classification (2010). Primary 47B20; Secondary 47B37. Keywords. Quasinormal operator, paranormal operator, spectral measure, weighted shift on a directed tree. 1. Introduction The class of bounded quasinormal operators was introduced by Brown in [3]. Two different definitions of unbounded quasinormal operators appeared inde- pendently in [15] and (a few years later) in [21]. In the present paper we show that both of these definitions coincide (cf. Theorem 3.1). We also discuss the question of whether the equality in Kaufman’s definition of quasinormality can be replaced by inclusion. It is shown that the answer is in the affirmative if the inclusion is properly chosen (cf. Theorem 3.1 and Remark 3.2). Next, we characterize quasinormality of unbounded operators in terms of the trun- cated operator Stieltjes moment problem [cf. Theorem 3.6(iv)].
    [Show full text]
  • Lecture Notes on Operator Theory
    1 Graduate Texts Woo Young Lee Lecture Notes on Operator Theory Spring 2008 Seoul National University Seoul, Korea Copyright c 2008 by the Seoul National University 2 . Lecture Notes on Operator Theory Woo Young Lee 3 . Preface The present lectures are based on a graduate course delivered by the author at the Seoul National University, in the spring semester of 2008. In these lectures I attempt to set forth some of the recent developments that had taken place in Operator Theory. In particular, I focus on the Fredholm and Weyl theory, hyponormal and subnormal theory, weighted shift theory, Toeplitz theory, and the invariant subspace problem. Seoul February, 2008 The author 4 Contents 1 Fredholm Theory 7 1.1 Introduction ................................. 7 1.2 Preliminaries ................................ 8 1.3 Definitions and Examples ......................... 10 1.4 Operators with Closed Ranges ...................... 11 1.5 The Product of Fredholm Operators ................... 14 1.6 Perturbation Theorems .......................... 16 1.7 The Calkin Algebra ............................ 18 1.8 The Punctured Neighborhood Theorem . 23 1.9 The Riesz-Schauder (or Browder) Theory . 26 1.10 Essential Spectra .............................. 32 1.11 Spectral Mapping Theorems ........................ 33 1.12 The Continuity of Spectra ......................... 35 1.13 Comments and Problems ......................... 38 2 Weyl Theory 39 2.1 Introduction ................................. 39 2.2 Weyl’s Theorem .............................. 41 2.3 Spectral Mapping Theorem for the Weyl spectrum . 53 2.4 Perturbation Theorems .......................... 59 2.5 Comments and Problems ......................... 66 3 Hyponormal and Subnormal Theory 69 3.1 Hyponormal Operators ........................... 69 3.2 The Berger-Shaw Theorem ........................ 73 3.3 Subnormal Operators ........................... 78 3.4 p-Hyponormal Operators ......................... 89 3.5 Comments and Problems ........................
    [Show full text]
  • Dynamics of the Commutator Operator
    Dynamics of the Commutator Operator Clifford Gilmore Department of Mathematics and Statistics University of Helsinki May 2014 HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department Science Mathematics and Statistics Tekijä — Författare — Author Clifford Gilmore Työn nimi — Arbetets titel — Title Dynamics of the Commutator Operator Oppiaine — Läroämne — Subject Mathematics Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Licentiate May 2014 79 Tiivistelmä — Referat — Abstract Linear dynamics is a rapidly evolving area of operator theory, however the only results related to the dynamics of the commutator operators have hitherto been on the characterisation of the hypercyclicity of the left and right multiplication operators. This text introduces the requisite background theory of hypercyclicity before surveying the hy- percyclicity of the left and right multiplication operators. It expands on this to prove sufficient conditions for the hypercyclicity of the two-sided multiplication operator. Conditions are established under which the general class of elementary operators are never hy- percyclic on Banach algebras and notably it is shown that elementary operators are never hy- percyclic on the space of bounded linear operators L (XAH ), where XAH is the Argyros–Haydon Banach space. In the first main result of this text, large classes of operators for which the induced commutator operators are never hypercyclic on separable Banach ideals are identified. In particular it is pro- ven that commutator operators induced by compact and Riesz operators are never hypercyclic on the ideal of compact operators and that commutator operators are also never hypercyclic on the ideal of compact operators K (XAH ).
    [Show full text]
  • Arxiv:1609.04986V1 [Math.FA] 16 Sep 2016 Hypercyclic for N Ih Utpiainoperators
    HYPERCYCLICITY PROPERTIES OF COMMUTATOR MAPS CLIFFORD GILMORE, EERO SAKSMAN, AND HANS-OLAV TYLLI Abstract. We investigate the hypercyclic properties of commutator maps acting on separable ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the space of compact operators on ℓ2. We also establish some necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps. 1. Introduction Let X be a Banach space and L (X) the space of bounded linear operators on X. The commutator operator ∆T : L (X) → L (X) induced by a fixed bounded linear operator T ∈ L (X) is defined as S 7→ ∆T (S)= TS − ST = LT (S) − RT (S) where S ∈ L (X) and LT ,RT : L (X) → L (X) are, respectively, the left and right multiplication operators. Recall for a separable Banach space X that the operator U ∈ L (X) is hypercyclic if there exists a vector x ∈ X (said to be a hypercyclic vector for U) such that its orbit under U is dense in X, that is {U nx : n ≥ 0} = X. The purpose of this paper is to initiate the investigation of hypercyclicity properties of commutator maps ∆T restricted to separable Banach ideals of operators, which turns out to be quite a subtle question. The motivation for this study is at least twofold. Firstly, Bonet et al. [4] arXiv:1609.04986v1 [math.FA] 16 Sep 2016 characterised the hypercyclicity of the left and right multiplication operators on such Banach ideals.
    [Show full text]
  • Generalized Paranormal Operators N
    Sivakumar N, Abirani Angelin S; International Journal of Advance Research, Ideas and Innovations in Technology. ISSN: 2454-132X Impact factor: 4.295 (Volume3, Issue2) Available online at www.ijariit.com Generalized Paranormal Operators N. Sivakumar S. Angelin Abirani Hindusthan College of Arts and Science, Tamil Nadu Hindusthan College of Arts and Science, Tamil Nadu [email protected] [email protected] Abstract: AN operator T ∈B (H) is said to be generalized P-paranormal if ퟏ ‖|푻|풑푼 |푻|풑풙 ‖‖풙‖ ≥ ‖|푻|풑풙‖ퟐ 푴풑 For all 풙 ∈ 푯, p > 0, and M > 0, where U is the partial isometry appeared in the polar decomposition 푻 = 푼|푻| of T. The aim of this note is to obtain some structure theorem for a class of generalized P-paranormal operators. Exactly we will give some conditions which are generalization of concepts of generalized paranormal operators. Keywords: Paranormal Operators, Hilbert Space, Hypo Normal Operator, Log-Hypo Normal Operators, and Bounded Linear Operator. INTRODUCTION Let H be an infinite dimensional complex Hilbert and B (H) denote the algebra of all bounded linear operators acting on H. Every operator T can be decomposed into T = U|푇|with a partial isometry U, where |푇| is the square root of T * T. If U is determined uniquely by the kernel condition ker (U) = ker|푇|, then this decomposition is called the polar decomposition, which is one of the most important results in operator theory. In this paper, T = U|푇| denotes the polar decomposition satisfying the kernel condition ker (U) = ker(|푇|). Recall that an operator T ∈B (H) is positive, T ≥ 0, if (T푥, 푥) ≥ 0 for all 푥 ∈ 퐻.
    [Show full text]
  • On Weyl Spectrum and Quasi-Similarity of Operators In
    ON WEYL SPECTRUM AND QUASI-SIMILARITY OF OPERATORS IN HILBERT SPACES by Peter Oyoo Maina A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Pure Mathematics School of Mathematics, University of Nairobi Kenya September, 2017 2 I Declaration and Approval I the undersigned declare that this thesis is a presentation of my original research work. Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any other university or any other institution of learning. PETER OYOO MAINA I80/80097/2008 Signature Date In our capacity as supervisors of the candidate‘s thesis, we certify that the above statements are true to the best of our knowledge and this thesis has our approval for submission. Prof. Jairus Khalagai School of Mathematics, University of Nairobi Signature Date Prof. Ganesh Phokariyal School of Mathematics, University of Nairobi Signature Date II Acknowledgment Am grateful and humbled to my God as the gardian angel in my academic domain, His blessings and insight has been wonderful throughout this thesis. My deepest gratitude to my first supervisor Prof. Jairus Khalagai for his inspiration, advice and guidance throughout the study and reading through the work. I would also like to thank the second supervisor Prof. Pokhariyal for his continuous value addition. It is through their patience, faith and constant encouragement that this research has been successfully concluded. They have provided the linearity to the kink in my life and the existing obstacles were manageable to manouver.
    [Show full text]
  • Some Properties of Quasinormal, Paranormal and $2-K
    The Australian Journal of Mathematical Analysis and Applications AJMAA Volume 14, Issue 2, Article 9, pp. 1-9, 2017 SOME PROPERTIES OF QUASINORMAL, PARANORMAL AND 2 − k∗ PARANORMAL OPERATORS SHQIPE LOHAJ Received 29 April, 2017; accepted 29 August, 2017; published 13 September, 2017. DEPARTMENT OF MATHEMATICS,UNIVERSITY OF PRISTINA, 10 000 KOSOVA [email protected] ABSTRACT. In the beginning of this paper some conditions under which an operator is par- tial isometry are given. Further, the class of 2 − k∗ paranormal operators is defined and some properties of this class in Hilbert space are shown. It has been proved that an unitarily operator equivalent with an operator of a 2 − k∗ paranormal operator is a 2 − k∗ paranormal operator, and if is a 2 − k∗ paranormal operator, that commutes with an isometric operator, then their product also is a 2 − k∗ paranormal operator. Key words and phrases: Quasinormal operators; 2 − k∗ paranormal operators. 2000 Mathematics Subject Classification. 47B20. ISSN (electronic): 1449-5910 c 2017 Austral Internet Publishing. All rights reserved. 2 SHQIPE LOHAJ 1. INTRODUCTION We will denote with H the Hilbert space and with L(H) the space of all bounded linear operators defined in Hilbert space H. The operator T ∈ L(H) is said to be paranormal, or an operator from (N) class, if k T x k2 ≤k T 2x k, (x ∈ H, k x k= 1), and ∗ paranormal if, k T ∗x k2 ≤k T 2x k, (x ∈ H, k x k= 1). The operator T ∈ L(H) is called k−paranormal or from (N; k) class, if k T x kk ≤k T kx k, (x ∈ H, k x k= 1) and T ∈ L(H) is called k∗−paranormal or from (N; k∗) class, if k T ∗x kk ≤k T kx k, (x ∈ H, k x k= 1).
    [Show full text]
  • A Study of Paranormal Operators on Hilbert Spaces
    International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.5, No.8, August 2016 DOI:10.15693/ijaist/2016.v5i8.69-74 A study of paranormal operators on Hilbert Spaces Mr. P.J.Dowari Dr.N.Goswami Research Scholar/Department of Mathematics Asstt Professor/Department of Mathematics Tripura University, Agartala, India GauhatiUniversity, Guwahati, India Abstract- In this paper we have studied bounded bilinear functionals from 푋∗ × 푌∗ to 핂,and paranormal and n-paranormal operators on is defined by Hilbert spaces and their tensor product. (푥 ⊗ 푦)(푓, 푔) = 푓(푥)푔(푦) (푓 ∈ 푋∗, 푔 ∈ 푌∗) Considering two paranormal operators on a The algebraic tensor product of X and Y, 푋 ⊗ 푌is Hilbert space푯, we derive the condition under defined to be linear span of which their different operations are again {푥 ⊗ 푦 ∶ 푥 ∈ 푋, 푦 ∈ 푌 } 푖푛 퐵퐿(푋∗, 푌∗; 핂). (refer paranormal. Also, taking two Hilbert spaces 푿 to [4]) and 풀 and a paranormal operator 푻ퟏ on 푿 and 푻ퟐ on 풀 we define a paranormal operator 푻 on their Definition 2.Given normed spaces X and Y, the projective tensor product. Different properties of projective tensor norm 훾on 푋 ⊗ 푌is defined by this operator 푻 regarding paranormality are also discussed here. 훾 푢 = inf ∥ 푥푖 ∥∥ 푦푖 ∥: 푢 = 푥푖 ⊗ 푦푖 푖 푖 where the infimum is taken over all finite AMS subject classification: 47A05, 47A07, 47A10, representations of u. 47A12, 47A15 The completion of 푋 ⊗ 푌with respect to 훾is called Keywords: algebraic tensor product, projective the projective tensor product of 푋and 푌and is tensor norm, paranormal, 푀-paranormal, 푛- denoted by 푋 ⊗ 푌.
    [Show full text]