Bibliography. Matrix Spaces and Schur Multipliers Downloaded from by on 01/18/18. For personal use only.

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1 Bibliography [1] J. Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces, Int. Eq. Oper. Th. 1, (1978), [2] J. Arazy, On the geometry of the unit ball of unitary matrix spaces, Int. Eq. Oper. Th., 4 (1981), [3] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math.,270 (1974), [4] J. M. Anderson and A. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc., 224 (1976), [5] J.Arazy,S.D.FisherandJ.Peetre,Möbius invariant function spaces, J. Reine Angew. Math. 363(1985), [6] A. B. Alexandrov, Essays on non locally convex Hardy classes, Lecture Notes in Math. 864 (1981), 1-89, Springer-Verlag, Berlin-Heidelberg-New York. [7] A. B. Alexandrov, On the boundary decay in the mean of harmonic functions, St. Petersburg Math. J. 5 (1996), [8] A. B. Alexandrov and V. V. Peller, Hankel and Toeplitz-Schur multipliers, Math. Ann., 324 (2002), [9] W. Arveson, Subalgebras of C algebras, Acta Math. 123 (1969), [10] W. Arveson, Interpolation problems in nest algebras, J. Funct. Analysis 20 (1975), [11] G. Bennett, Schur multipliers, Duke Math. J., 44 (1977), [12] O. Blasco, A characterization of Hilbert spaces in terms of multipliers between spaces of vector-valued analytic functions, Michigan Math. J., 42(1995), [13] O. Blasco, Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces, Illinois J. Math., 41 (1997), [14] O. Blasco, Introduction to vector valued Bergman spaces, Function spaces and operator theory, 9 30, Univ. Joensuu Dept. Math. Rep. Ser., 8, Univ. Joensuu, Joensuu, [15] O. Blasco and A. Pelczynski, Theorems of Hardy and Paley for vector valued analytic functions and related classes of Banach spaces, Trans. Amer. Math. Soc. 323 (1991),

2 186 Matrix spaces and Schur multipliers: Matriceal harmonic analysis [16] M. Bozejko, Littlewood functions, Hankel multipliers and power bounded operators on a Hilbert space, Colloquium Math. 51 (1987), [17] S. Barza, L. E. Persson and N. Popa, A matriceal analogue of Fejer s theory, Math. Nachr.,260 (2003), [18] S. Barza, V. D. Lie and N. Popa, Approximation of infinite matrices by matriceal Haar polynomials,ark. Mat. 43 (2005), no. 2, [19] L. Bergh and J. Lofstrom, Interpolation spaces. An Introduction, Springer Verlag, Berlin, [20] S. Barza, D. Kravvaritis and N. Popa, Matriceal Lebesgue spaces and Hoelder inequality, J. Funct. Spaces Appl. 3 (2005), no. 3, [21] R. R. Coifman, A real variable characterization of H p, Studia Math., 51 (1974), [22] R. G. Cooke, Infinite matrices and sequence spaces, Macmillan [23] P. L. Duren, Theory of H p Spaces, Academic Press, [24] R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, [25] M. Déchamps-Godim, F. Lust-Piquard and H. Queffelec, On the minorant properties in C p(h), Pacific J. Math. 119 (1985), [26] P. Dodds and F. Sukochev, RUC-decompositions in symmetric operator spaces, Integr. Equat. Oper. Th., 29(1997), [27] P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on H p spaces with 0 <p<1, J. Reine Angew. Math., 238 (1969), [28] P. L. Duren and A. Schuster, Bergman spaces, American Mathematical Society, Mathematical Surveys and Monographs, vol. 100, Providence, [29] P. L. Duren and A. L. Shields, Properties of H p (0 <p<1) and its containing Banach space, Trans. Amer. Math. Soc., 141 (1969), [30] R. E. Edwards, Functional Analysis; Theory and Applications, Holt, Rinehart and Winston, New-York, [31] L. Fejér, Untersuchungen über Fouriersche Reihen, Math. Ann., 58 (1904), [32] C. L. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math., 129(1972), [33] J. B. Garnett, Bounded analytic functions, Academic Press, New York, [34] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I [35] A. Grothendieck, Resumé delathéorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. Sao Paulo, 8 (1956), [36] W. W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), [37] A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann., 69, (1910), [38] A. Harcharras, Fourier analysis, Schur multipliers on S p and noncommutative Λ(p)-sets, Studia Math., 137,(3), (1999), [39] G. H. Hardy and J. E. Littlewood, Notes on the theory of series (XX) Gen-

3 Bibliography 187 eralizationsofatheoremofpaley,quart. J. Math., Oxford Ser., 8 (1937), [40] G. H. Hardy and J. E. Littlewood, A new proof of a theorem on rearrangements, J. London Math. Soc. 23 (1948), [41] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, [42] J. R. Holub, On the metric geometry of Ideals of Operators on Hilbert Space, Math. Ann. 201 (1973), [43] C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 41(1974), [44] C. Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), [45] U. Haagerup and G. Pisier, Factorization of analytic functions with values in non-commutative L 1 spaces and applications, Can. J. Math.,XLI, (1989), [46] D. R Jocic and D. Krtinic, Schur-Laurent multipliers for block matrices and geometric characterization of continuous matrices, Linear and Multilinear Algebra, vol. 58,(2010), [47] M. Jevtic and M. Pavlovic, Coefficient multipliers on spaces of analytic functions, Acta Sci. Math. (Szeged), 64 (1998), [48] D. Krtinic, A matricial analogue of Fejer s theory for different types of convergence, Math. Nachr., 280 (2007), 1-6. [49] S. Kwapien and A. Pelczynski, Themaintriangleprojectioninmatrixspaces and its applications, Studia Math.,34,(1970), [50] V. Lie, Integral operators in infinite matrix theory; the study of matriceal Riesz projection, (Romanian), Master s dissertation at University of Bucharest, 49p, [51] F. Lust-Piquard, Inégalités de Khintchine dans C p (1 < p < ), C. R. Acad. Sc. Paris, 303,Série I, (1986), [52] F. Lust-Piquard, On the coefficient problem: a version of the Kahane Katznelson de Leeuw theorem for spaces of matrices, J. Funct. Anal., 149 (1997), [53] F. Lust-Piquard and G. Pisier, Non-commutative Khintchine and Paley inequalities, Ark. Mat. (29)(1991), [54] J. Lindenstrauss and H. P. Rosenthal, The L p spaces, Israel J. Math. 7 (1969), [55] A. Marcoci, Some new results concerning Lorentz sequence spaces and Schur multipliers, Licentiate thesis, Lule a University of Technology, Sweden, [56] L. Marcoci, Some new results concerning Schur multipliers and duality results between Bergman-Schatten and little Bloch spaces, Licentiate thesis, Lule a University of Technology, Sweden, [57] A. Marcoci, Some new results concerning Banach spaces of infinite matrices and Lorentz sequence spaces, Doctoral thesis, Lule a University of Technology, Sweden, [58] L. Marcoci, A study of Schur multipliers and some Banach spaces of infinite

4 188 Matrix spaces and Schur multipliers: Matriceal harmonic analysis matrices, Doctoral thesis, Lule a University of Technology, Sweden, [59] M. Marsalli, Noncommutative H 2 spaces, Proc. Amer. Math. Soc., 125 (1997), [60] M. Mateljevic and M. Pavlovic, Multipliers of H p and BMOA, Pacific J. Math. 146 (1990), [61] M. Mateljevic and M. Pavlovic, L p behaviour of the integral means of analytic functions, Studia Math. 77, (1984), [62] L.G. Marcoci, L.E. Persson, I. Popa and N. Popa, A new characterization of BergmanSchatten spaces and a duality result, J. Math. Anal. Appl, 360 (2009), [63] N. Nikolskii, Treatise on the shift operator, Springer Verlag, Berlin, [64] D. M. Oberlin, Translation-invariant operators on L p (G), 0 <p<1, Michigan Math. J. 23 (1976), [65] V. Peller, Hankel Operators and Their Applications, Springer- Verlag, New York, [66] V. Peller, A description of Hankel operators of class S p for p>0, an investigation of the rate of rational approximation and other applications, Mat. Sbornik, 122 (1983), [67] V. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math. 146, Longman, Wiley, New-York, [68] M. Pavlović, Introduction to function spaces on the disk. Matematicki Institut SANU, Beograd, [69] L. B. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc., 150 (1970), [70] S. Parrott, On a quotient norm and the Sz.-Nagy-Foiaş lifting theorem,j. Funct. Anal., 30 (1978), [71] J. Peetre, New thoughts on Besov spaces, Duke Univ. Press., Durham, N.C., [72] N. Popa, Matriceal Bloch and Bergman-Schatten spaces, Rev. Roumaine Math. Pures Appl., 52 (2007), [73] N. Popa, A characterization of upper triangular trace class matrices, C. R. Acad. Sci. Paris, Ser. I 347 (2009), [74] N. Popa, Schur multipliers between Banach spaces of matrices, Proceedings of the Sixth Congress of Romanian Mathematicians Bucharest, 2007, vol. 1, [75] I. Popa and N. Popa Matrices of bounded variation, in Proceeding of the International Conference: Mathematical Analysis and its Applications, Athens 2002, pp [76] I. Popa and N. Popa Inequalities in matrix spaces, Math. Rep., 12(62) (2010), [77] S. Power, Commutators with the triangular projection and Hankel forms on nest algebras, J. London Math. Soc. 32 (1985), [78] I. I. Privalov, Randeigenschaften analytischer Funktionen, Verlag der Wiss., Berlin, [79] A. Pelczynski and F. Sukochev, Some remarks on Toeplitz multipliers and

5 Bibliography 189 Hankel matrices, Studia Math., 175, (2006), [80] G. Pisier, Multipliers of the Hardy space H 1 and power bounded operators, Preprint, [81] G. Pisier, Similarity problems and completely bounded maps, LNM 1618, Springer Verlag, Berlin, [82] A. Pietsch, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin, [83] A. Pietsch and H. Triebel, Interpolationstheorie für Banachideale von beschränkten linearen Operatoren, Studia Math., 31,(1968), [84] A. L. Shields, An analogue of a Hardy-Littlewood-Fejer inequality for upper triangular trace class operators, Math. Z., 182 (1983), [85] D. Sarason, Generalized interpolation in H, Trans. Amer. Math. Soc., 127 (1967), [86] I. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math., 140 (1911), [87] B. Simon, Trace ideals and their applications, London Math. Soc Lecture Notes Series, no. 35, Cambridge, [88] W. T. Sledd and D. A. Stegenga, An H 1 multiplier theorem, Ark. Mat., 19 (1981), no. 2, [89] B. Smith, A strong convergence theorem for H 1 (T), Banach spaces, harmonic analysis and probability theory, (Storrs, Conn., 1980/1981), Lecture Notes in Math., vol. 995, Springer-Verlag, Berlin, 1983, [90] E. M. Stein and R. Shakarchi, Fourier analysis; An Introduction, Princeton University Press, Princeton, [91] W. Stinespring, Positive functions on C -algebras, Proc. Amer. Math. Soc., 6 (1966), [92] A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162 (1971), [93] H. Triebel, Interpolation theory, function spaces, differential operators. Second edition. Johann Ambrosius Barth, Heidelberg, [94] K. Zhu, Operator theory in Banach function spaces, Marcel Dekker, New- York, [95] K. Zhu, Analytic Besov space, J. Math. Anal. Appl., 157, (1991), [96] A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, [97] N. Wiener, The quadratic variation of a function and its Fourier coefficients, Massachusett s J. Math., 3 (1924), [98] G. Wittstock, Ein Operatorenwertigen Hahn-Banach Satz, J. Funct. Anal., 40 (1981),

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7 Index (H 1 l 1)-multiplier, 80 (X, Y ) the space of all Schur multipliers from X into Y,1 (, ) the scalar product in a Hilbert space, 3 (p, q)-bounded, 80 A B the Schur product of the matrices A and B, 1 A k the kth-diagonal matrix, 1 B(l 2) the space of all bounded linear operators on l 2,2 BMOA the space of analytic functions of bounded mean oscillation, 4 BMOA(l 2), 109 BMO F (l 2), BMOA F (l 2), xi, 114 C 1 the trace class, 3 C 2 the Hilbert-Schmidt class, 3 C p, 0 <p< Schatten class, 3 D A the defect operator, 92 E t the Toeplitz matrix with entries (e i(j k)t ) j,k 1,71 FM,84 H 2 (H) Hardy space of Hilbert space-valued functions, 112 H,,1,H,,1 (l 2), 176 H (l 2), 114 H p (l 2) the matriceal Hardy space of index p, 67 H p, 0 <p Hardy spaces, 4 H p X vector-valued Hardy space, 79 H 1,,1 1 (l 2), 176 H Φ matrix version of Hankel operator, 113 L 1 (D, l 2), L (D, l 2), 5 L p,unc a (D, l 2), 132 M(l 2) the space of all Schur multipliers on B(l 2), 2 M(T) the convolution algebra of Borel measures on T, 101 P, P Bergman projection, 128 P T the triangular projection, 69 P α, 129 T p(l 2 R), Tp(l 2 C) spaces of upper triangular matrices, 70 T p, 0 <p< the matrix analogue of Hardy space H p,69 VMO F (l 2), 118 Γ Ω the block Hankel matrix, 96 β(z, w) the Bergman metric, 7 l (l ), l 2(l 2,w), 134 B Bloch space, 4 B(D, l 2) the matriceal Bloch space, 150 B 0,c(D, l 2), 170 B 0 the little Bloch space, 4 B 0(D, l 2) little Bloch space, 161 C(D, l 2 ), 165 C 0(D, C ), 171 C 0(D, l 2), 169 C 0 ɛ C ɛ-tensor product of corresponding Banach spaces, 171 I, 152 L A k, L A k, L A(x, t),

8 192 Matrix spaces and Schur multipliers: Matriceal harmonic analysis M p,2, 119 T,2 ρ(z, w) the pseudo-hyperbolic distance, 7 σ n(a), 137 f A(r, t) = k= A k(r)e ikt,2 m X the operator induced by the multiplier m, 80 w -measurable function, 5 VMOA(l 2), 109 Bergman-Schatten classes, 122 Analytic matrices, 2 Banach space of (H 1 l 1)-Fourier type, 81 Bennett s Theorem, 2 Bloch matrix, 150 crudely finitely representable Banach space, 86 Matriceal Hausdorff-Young Theorem, 75 Matrix version of Nehari theorem, xi, 114 Pavlović Theorem,97 Schmidt Theorem, 3 Shields inequality, 70 singular values of an operator T,3 strongly measurable function, 5 the Hankel matrix H a,90 The matrix spaces L p r(l 2), resp. L p c(l 2), 1 p 2, 66 The noncommutative factorization theorem, 87 the projective tensor product, 88 Toeplitz matrices, 2 Vectorial Nehari theorem, 113