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Metric, Schauder and Operator-Valued Frames

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Contributed by K. MAHESH KRISHNA ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. It is proved that every separable metric space admits a metric Md\mathcal{M}_d-frame. Through Lipschitz-free Banach spaces it is showed that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. Several characterizations of metric frames are obtained. Stability results are also presented. Non linear multipliers are introduced and studied. This notion is connected with the notion of Lipschitz compact operators. Continuity properties of multipliers are discussed. For a subclass of approximated Schauder frames for Banach spaces, characterization result is derived using standard Schauder basis for standard sequence spaces. Duals of a subclass of approximate Schauder frames are completely described. Similarity of this class is characterized and interpolation result is derived using orthogonality. A dilation result is obtained. A new identity is derived for Banach spaces which admit a homogeneous semi-inner product. Some stability results are obtained for this class. A generalization of operator-valued frames for Hilbert spaces are introduced which unifies all the known generalizations of frames for Hilbert spaces. This notion has been studied in depth by imposing factorization property of the frame operator. Its duality, similarity and orthogonality are addressed. Connections between this notion and unitary representations of groups and group-like unitary systems are derived. Paley-Wiener theorem for this class are derived.

References

  1. Achour, D., Rueda, P., Sánchez-Pérez, E. A., and Yahi, R. (2016). Lipschitz operator ideals and the approximation property. J. Math. Anal. Appl. , 436(1):217--236.
  2. Adcock, B. and Huybrechs, D. (2019). Frames and numerical approximation. SIAM Rev. , 61(3):443--473.
  3. Agler, J. and McCarthy, J. E. (2002). Pick interpolation and H ilbert function spaces , volume 44 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI.
  4. Aharoni, I. (1974). Every separable metric space is L ipschitz equivalent to a subset of c0+c^{+}_{0} . Israel J. Math. , 19:284--291.
  5. Akcoglu, M. A. and Sucheston, L. (1977). Dilations of positive contractions on LpL_{p} spaces. Canad. Math. Bull. , 20(3):285--292.
  6. Albiac, F. and Kalton, N. J. (2016). Topics in B anach space theory , volume 233 of Graduate Texts in Mathematics . Springer.
  7. Aldroubi, A., Baskakov, A., and Krishtal, I. (2008). Slanted matrices, B anach frames, and sampling. J. Funct. Anal. , 255(7):1667--1691.
  8. Aldroubi, A., Cabrelli, C., Cakmak, A. F., Molter, U., and Petrosyan, A. (2017a). Iterative actions of normal operators. J. Funct. Anal. , 272(3):1121--1146.
  9. Aldroubi, A., Cabrelli, C., Molter, U., and Tang, S. (2017b). Dynamical sampling. Appl. Comput. Harmon. Anal. , 42(3):378--401.
  10. Aldroubi, A., Cabrelli, C., and Molter, U. M. (2004). Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for L2(Rd)L^2({\Bbb R}^d) . Appl. Comput. Harmon. Anal. , 17(2):119--140.
Show all 221 referencesShow fewer references
  1. Aldroubi, A. and Petrosyan, A. (2017). Dynamical sampling and systems from iterative actions of operators. In Frames and other bases in abstract and function spaces , Appl. Numer. Harmon. Anal., pages 15--26. Birkhäuser/Springer.
  2. Aldroubi, A., Sun, Q., and Tang, W.-S. (2001). pp -frames and shift invariant subspaces of LpL^p . J. Fourier Anal. Appl. , 7(1):1--21.
  3. Allan, G. R. (2011). Introduction to B anach spaces and algebras , volume 20. Oxford University Press, Oxford.
  4. Ambrozie, C. and Muller, V. (2014). Commutative dilation theory,. In Operator Theory , pages 1--29. Springer, Basel.
  5. Ando, T. (1963). On a pair of commutative contractions. Acta Sci. Math. (Szeged) , 24:88--90.
  6. Arens, R. F. and Eells, Jr., J. (1956). On embedding uniform and topological spaces. Pacific J. Math. , 6:397--403.
  7. Arveson, W. (2010). Dilation theory yesterday and today. In A glimpse at H ilbert space operators , volume 207 of Oper. Theory Adv. Appl. , pages 99--123. Birkhäuser Verlag, Basel.
  8. Bachman, G., Narici, L., and Beckenstein, E. (2000). Fourier and wavelet analysis . Universitext. Springer-Verlag, New York.
  9. Bakic, D. and Beric, T. (2015). On excesses of frames. Glas. Mat. Ser. III , 50(70)(2):415--427.
  10. Balan, R. (1999). Equivalence relations and distances between H ilbert frames. Proc. Amer. Math. Soc. , 127(8):2353--2366.
  11. Balan, R., Casazza, P., and Edidin, D. (2006). On signal reconstruction without phase. Appl. Comput. Harmon. Anal. , 20(3):345--356.
  12. Balan, R., Casazza, P. G., Edidin, D., and Kutyniok, G. (2005). Decompositions of frames and a new frame identity . Wavelets XI, SPIE Proc. 5914, San Diego, California, United States.
  13. Balan, R., Casazza, P. G., Edidin, D., and Kutyniok, G. (2007). A new identity for P arseval frames. Proc. Amer. Math. Soc. , 135(4):1007--1015.
  14. Balan, R., Casazza, P. G., Heil, C., and Landau, Z. (2003). Deficits and excesses of frames. Adv. Comput. Math. , 18(2-4):93--116.
  15. Balan, R. V. (1998). A study of W eyl- H eisenberg and wavelet frames . ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)--Princeton University.
  16. Balazs, P. (2007). Basic definition and properties of B essel multipliers. J. Math. Anal. Appl. , 325(1):571--585.
  17. Balazs, P., Antoine, J.-P., and Grybos, A. (2010). Weighted and controlled frames: mutual relationship and first numerical properties. Int. J. Wavelets Multiresolut. Inf. Process. , 8(1):109--132.
  18. Balazs, P., Holighaus, N., Necciari, T., and Stoeva, D. (2017). Frame theory for signal processing in psychoacoustics. In Excursions in harmonic analysis. V ol. 5 , Appl. Numer. Harmon. Anal., pages 225--268. Birkhäuser/Springer, Cham.
  19. Banaś, J. and Mursaleen, M. (2014). Sequence spaces and measures of noncompactness with applications to differential and integral equations . Springer, New Delhi.
  20. Benedetto, J. J. and Ferreira, P. J. S. G., editors (2001). Modern sampling theory: Mathematics and applications . Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA.
  21. Benedetto, J. J. and Fickus, M. (2003). Finite normalized tight frames. Adv. Comput. Math. , 18(2-4):357--385.
  22. Benedetto, J. J. and Kolesar, J. D. (2006). Geometric properties of G rassmannian frames for R2\mathbb{R}^2 and R3\mathbb{R}^3 . EURASIP Journal on Applied Signal Processing , pages 1--17.
  23. Benedetto, J. J. and Pfander, G. E. (2006). Frame expansions for G abor multipliers. Appl. Comput. Harmon. Anal. , 20(1):26--40.
  24. Benedetto, J. J., Powell, A. M., and Y lmaz, O. (2006). Sigma- D elta ( ΣΔ\Sigma\Delta ) quantization and finite frames. IEEE Trans. Inform. Theory , 52(5):1990--2005.
  25. Berge, E. (2022). A primer on coorbit theory. J. Fourier Anal. Appl. , 28(1):Paper No. 2, 61.
  26. Bhat, B. V. R., De, S., and Rakshit, N. (2021). A caricature of dilation theory. Adv. Oper. Theory , 6(4):Paper No. 63, 20.
  27. Bhat, B. V. R. and Mukherjee, M. (2020). Two states. To appear in Houston Journal of Mathematics, arXiv:1710.00180v3 [math.FA] .arXiv
  28. Bhattacharyya, T. (2002). Dilation of contractive tuples: a survey. In Surveys in analysis and operator theory ( C anberra, 2001) , volume 40 of Proc. Centre Math. Appl. Austral. Nat. Univ. , pages 89--126. Austral. Nat. Univ., Canberra.
  29. Blanchard, P. and Bruning, E. (2003). Hilbert spaces: A brief historical introduction. In Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Methods , pages 173--183. Birkh ä user Boston.
  30. Bodmann, B. G. and Haas, J. I. (2020). A short history of frames and quantum designs. In Topological phases of matter and quantum computation , volume 747 of Contemp. Math. , pages 215--226. Amer. Math. Soc., Providence, RI.
  31. Bodmann, B. G. and Paulsen, V. I. (2005). Frames, graphs and erasures. Linear Algebra Appl. , 404:118--146.
  32. Borup, L. and Nielsen, M. (2007). Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. , 13(1):39--70.
  33. Brown, A. and Pearcy, C. (1965). Structure of commutators of operators. Ann. of Math. (2) , 82:112--127.
  34. Carando, D., Lassalle, S., and Schmidberg, P. (2011). The reconstruction formula for B anach frames and duality. J. Approx. Theory , 163(5):640--651.
  35. Casazza, P., Christensen, O., and Stoeva, D. T. (2005a). Frame expansions in separable B anach spaces. J. Math. Anal. Appl. , 307(2):710--723.
  36. Casazza, P. G. (1998). Every frame is a sum of three (but not two) orthonormal bases---and other frame representations. J. Fourier Anal. Appl. , 4(6):727--732.
  37. Casazza, P. G. (2001). Approximation properties. In Handbook of the geometry of B anach spaces, V ol. I , pages 271--316. North-Holland, Amsterdam.
  38. Casazza, P. G. and Christensen, O. (1997). Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. , 3(5):543--557.
  39. Casazza, P. G. and Christensen, O. (2008). The reconstruction property in B anach spaces and a perturbation theorem. Canad. Math. Bull. , 51(3):348--358.
  40. Casazza, P. G., Christensen, O., Lindner, A. M., and Vershynin, R. (2005b). Frames and the F eichtinger conjecture. Proc. Amer. Math. Soc. , 133(4):1025--1033.
  41. Casazza, P. G., Dilworth, S. J., Odell, E., Schlumprecht, T., and Zsak, A. (2008a). Coefficient quantization for frames in B anach spaces. J. Math. Anal. Appl. , 348(1):66--86.
  42. Casazza, P. G. and Edidin, D. (2007). Equivalents of the K adison- S inger problem. In Function spaces , volume 435 of Contemp. Math. , pages 123--142. Amer. Math. Soc., Providence, RI.
  43. Casazza, P. G., Han, D., and Larson, D. R. (1999). Frames for B anach spaces. In The functional and harmonic analysis of wavelets and frames ( S an A ntonio, TX , 1999) , volume 247 of Contemp. Math. , pages 149--182. Amer. Math. Soc., Providence, RI.
  44. Casazza, P. G. and Kalton, N. J. (1999). Generalizing the P aley- W iener perturbation theory for B anach spaces. Proc. Amer. Math. Soc. , 127(2):519--527.
  45. Casazza, P. G. and Kutyniok, G. (2004). Frames of subspaces. In Wavelets, frames and operator theory , volume 345 of Contemp. Math. , pages 87--113. Amer. Math. Soc., Providence, RI.
  46. Casazza, P. G. and Kutyniok, G., editors (2013). Finite frames: Theory and applications . Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York.
  47. Casazza, P. G., Kutyniok, G., and Li, S. (2008b). Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. , 25(1):114--132.
  48. Casazza, P. G. and Tremain, J. C. (2016). Consequences of the M arcus/ S pielman/ S rivastava solution of the K adison- S inger problem. In New trends in applied harmonic analysis , Appl. Numer. Harmon. Anal., pages 191--213. Birkhäuser/Springer, Cham.
  49. Chen, D. and Zheng, B. (2012). Lipschitz pp -integral operators and L ipschitz pp -nuclear operators. Nonlinear Anal. , 75(13):5270--5282.
  50. Christensen, J. G., Gröchenig, K., and Ólafsson, G. (2017). New atomic decompositions for B ergman spaces on the unit ball. Indiana Univ. Math. J. , 66(1):205--235.
  51. Christensen, O. (1995a). Frame perturbations. Proc. Amer. Math. Soc. , 123(4):1217--1220.
  52. Christensen, O. (1995b). Frames and pseudo-inverses. J. Math. Anal. Appl. , 195(2):401--414.
  53. Christensen, O. (1995c). A P aley- W iener theorem for frames. Proc. Amer. Math. Soc. , 123(7):2199--2201.
  54. Christensen, O. (2001). Frames, R iesz bases, and discrete G abor/wavelet expansions. Bull. Amer. Math. Soc. (N.S.) , 38(3):273--291.
  55. Christensen, O. (2016). An I ntroduction to F rames and R iesz bases . Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA.
  56. Christensen, O. and Eldar, Y. C. (2004). Oblique dual frames and shift-invariant spaces. Appl. Comput. Harmon. Anal. , 17(1):48--68.
  57. Christensen, O., Hasannasab, M., and Rashidi, E. (2018). Dynamical sampling and frame representations with bounded operators. J. Math. Anal. Appl. , 463(2):634--644.
  58. Christensen, O. and Heil, C. (1997). Perturbations of B anach frames and atomic decompositions. Math. Nachr. , 185:33--47.
  59. Christensen, O. and Stoeva, D. T. (2003). pp -frames in separable B anach spaces. Adv. Comput. Math. , 18(2-4):117--126.
  60. Conway, J. B. (2000). A course in operator theory , volume 21 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI.
  61. Cordero, E., Gröchenig, K., and Nicola, F. (2012). Approximation of F ourier integral operators by G abor multipliers. J. Fourier Anal. Appl. , 18(4):661--684.
  62. Crabb, M. J. and Davie, A. M. (1975). von N eumann's inequality for H ilbert space operators. Bull. London Math. Soc. , 7:49--50.
  63. Cuntz, J. (1977). Simple $C\sp*$ -algebras generated by isometries. Comm. Math. Phys. , 57(2):173--185.
  64. Czaja, W. (2008). Remarks on N aimark's duality. Proc. Amer. Math. Soc. , 136(3):867--871.
  65. Dahlke, S., Fornasier, M., Rauhut, H., Steidl, G., and Teschke, G. (2008). Generalized coorbit theory, B anach frames, and the relation to α\alpha -modulation spaces. Proc. Lond. Math. Soc. (3) , 96(2):464--506.
  66. Dahlke, S., Steidl, G., and Teschke, G. (2004). Weighted coorbit spaces and B anach frames on homogeneous spaces. J. Fourier Anal. Appl. , 10(5):507--539.
  67. Dahlke, S., Steidl, G., and Teschke, G. (2007). Frames and coorbit theory on homogeneous spaces with a special guidance on the sphere. J. Fourier Anal. Appl. , 13(4):387--404.
  68. Dai, X.-R. and Sun, Q. (2016). The abcabc -problem for G abor systems. Mem. Amer. Math. Soc. , 244(1152):ix+99.
  69. Daubechies, I., Grossmann, A., and Meyer, Y. (1986). Painless nonorthogonal expansions. J. Math. Phys. , 27(5):1271--1283.
  70. Davidson, K. R. (1996). C∗C^* -algebras by example , volume 6 of Fields Institute Monographs . American Mathematical Society, Providence, RI.
  71. de Sz. Nagy, B. (1947). Expansion theorems of P aley- W iener type. Duke Math. J. , 14:975--978.
  72. Donoho, D. L. and Elad, M. (2003). Optimally sparse representation in general (nonorthogonal) dictionaries via l1l^1 minimization. Proc. Natl. Acad. Sci. USA , 100(5):2197--2202.
  73. Dorfler, M. and Torresani, B. (2010). Representation of operators in the time-frequency domain and generalized G abor multipliers. J. Fourier Anal. Appl. , 16(2):261--293.
  74. Dosev, D. and Johnson, W. B. (2010). Commutators on ℓ∞\ell_\infty . Bull. Lond. Math. Soc. , 42(1):155--169.
  75. Dosev, D., Johnson, W. B., and Schechtman, G. (2013). Commutators on LpL_p , 1≤p<∞1\leq p<\infty . J. Amer. Math. Soc. , 26(1):101--127.
  76. Dosev, D. T. (2009). Commutators on l1l_1 . J. Funct. Anal. , 256(11):3490--3509.
  77. Drury, S. W. (1983). Remarks on von N eumann's inequality. In Banach spaces, harmonic analysis, and probability theory ( S torrs, C onn., 1980/1981) , volume 995 of Lecture Notes in Math. , pages 14--32. Springer, Berlin.
  78. Dubei, M., Tymchatyn, E. D., and Zagorodnyuk, A. (2009). Free B anach spaces and extension of L ipschitz maps. Topology , 48(2-4):203--212.
  79. Duffin, R. J. and Schaeffer, A. C. (1952). A class of nonharmonic F ourier series. Trans. Amer. Math. Soc. , 72:341--366.
  80. Dykema, K., Figiel, T., Weiss, G., and Wodzicki, M. (2004). Commutator structure of operator ideals. Adv. Math. , 185(1):1--79.
  81. Dykema, K. and Skripka, A. (2012). On single commutators in II1{II}_1 -factors. Proc. Amer. Math. Soc. , 140(3):931--940.
  82. Egervary, E. (1954). On the contractive linear transformations of nn -dimensional vector space. Acta Sci. Math. (Szeged) , 15:178--182.
  83. Eldar, Y. C. and Forney, Jr., G. D. (2002). Optimal tight frames and quantum measurement. IEEE Trans. Inform. Theory , 48(3):599--610.
  84. Enflo, P. (1973). A counterexample to the approximation problem in B anach spaces. Acta Math. , 130:309--317.
  85. Fabian, M., Habala, P., Hájek, P., Montesinos, V., and Zizler, V. (2011). Banach space theory: The basis for linear and nonlinear analysis . CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York.
  86. Fackler, S. and Gluck, J. (2019). A toolkit for constructing dilations on B anach spaces. Proc. Lond. Math. Soc. (3) , 118(2):416--440.
  87. Favier, S. J. and Zalik, R. A. (1995). On the stability of frames and R iesz bases. Appl. Comput. Harmon. Anal. , 2(2):160--173.
  88. Feichtinger, H. G. (2006). Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process. , 5(2):109--140.
  89. Feichtinger, H. G. (2015). Choosing function spaces in harmonic analysis. In Excursions in harmonic analysis. V ol. 4 , Appl. Numer. Harmon. Anal., pages 65--101. Birkhäuser/Springer, Cham.
  90. Feichtinger, H. G. and Gröchenig, K. (1988). A unified approach to atomic decompositions via integrable group representations. In Function spaces and applications ( L und, 1986) , volume 1302 of Lecture Notes in Math. , pages 52--73. Springer, Berlin.
  91. Feichtinger, H. G. and Gröchenig, K. H. (1989a). Banach spaces related to integrable group representations and their atomic decompositions. I . J. Funct. Anal. , 86(2):307--340.
  92. Feichtinger, H. G. and Gröchenig, K. H. (1989b). Banach spaces related to integrable group representations and their atomic decompositions. II . Monatsh. Math. , 108(2-3):129--148.
  93. Feichtinger, H. G., Kozek, W., and Luef, F. (2009). Gabor analysis over finite A belian groups. Appl. Comput. Harmon. Anal. , 26(2):230--248.
  94. Feichtinger, H. G., Luef, F., and Werther, T. (2007). A guided tour from linear algebra to the foundations of G abor analysis. In Gabor and wavelet frames , volume 10 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. , pages 1--49. World Sci. Publ., Hackensack, NJ.
  95. Feichtinger, H. G. and Nowak, K. (2003). A first survey of G abor multipliers. In Advances in G abor analysis , Appl. Numer. Harmon. Anal., pages 99--128. Birkhäuser Boston, Boston, MA.
  96. Feichtinger, H. G. and Strohmer, T., editors (1998). Gabor A nalysis and A lgorithms : T heory and A pplications . Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA.
  97. Feichtinger, H. G. and Strohmer, T., editors (2003). Advances in G abor A nalysis . Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA.
  98. Fickus, M., Massar, M. L., and Mixon, D. G. (2013). Finite frames and filter banks. In Finite frames , Appl. Numer. Harmon. Anal., pages 337--379. Birkhäuser/Springer, New York.
  99. Fornasier, M. (2004). Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl. , 289(1):180--199.
  100. Fornasier, M. (2007). Banach frames for α\alpha -modulation spaces. Appl. Comput. Harmon. Anal. , 22(2):157--175.
  101. Freeman, D., Odell, E., Schlumprecht, T., and Zsák, A. (2014). Unconditional structures of translates for Lp(Rd)L_p(\mathbb{R}^d) . Israel J. Math. , 203(1):189--209.
  102. Gabardo, J.-P. and Han, D. (2001). Subspace W eyl- H eisenberg frames. J. Fourier Anal. Appl. , 7(4):419--433.
  103. Gabardo, J.-P. and Han, D. (2003). Frame representations for group-like unitary operator systems. J. Operator Theory , 49(2):223--244.
  104. Gabor, D. (1946). Theory of communication. J. IEEE , 93(26):429--457.
  105. Gavruta, P. (2006). On some identities and inequalities for frames in H ilbert spaces. J. Math. Anal. Appl. , 321(1):469--478.
  106. Gelfand, I. (1941). Normierte R inge. Rec. Math. [Mat. Sbornik] N. S. , 9 (51):3--24.
  107. Gelfand, I. and Neumark, M. (1943). On the imbedding of normed rings into the ring of operators in H ilbert space. Rec. Math. [Mat. Sbornik] N.S. , 12(54):197--213.
  108. Gibson, P. C., Lamoureux, M. P., and Margrave, G. F. (2013). Representation of linear operators by G abor multipliers. In Excursions in harmonic analysis. V olume 2 , Appl. Numer. Harmon. Anal., pages 229--250. Birkhäuser/Springer, New York.
  109. Giles, J. R. (1967). Classes of semi-inner-product spaces. Trans. Amer. Math. Soc. , 129:436--446.
  110. Godefroy, G. (2015). A survey on L ipschitz-free B anach spaces. Comment. Math. , 55(2):89--118.
  111. Gohberg, I. C. and Krein, M. G. (1969). Introduction to the theory of linear nonselfadjoint operators . American Mathematical Society, Providence, R.I.
  112. Gröchenig, K. (1991). Describing functions: atomic decompositions versus frames. Monatsh. Math. , 112(1):1--42.
  113. Gröchenig, K. (2001). Foundations of T ime- F requency A nalysis . Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA.
  114. Gröchenig, K. (2004). Localization of frames, B anach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. , 10(2):105--132.
  115. Halmos, P. R. (1950). Normal dilations and extensions of operators. Summa Brasil. Math. , 2:125--134.
  116. Halmos, P. R. (1982). A H ilbert space problem book , volume 19 of Graduate Texts in Mathematics . Springer-Verlag, New York-Berlin.
  117. Han, D. and Juste, T. (2019). Phase-retrievable operator-valued frames and representations of quantum channels. Linear Algebra Appl. , 579:148--168.
  118. Han, D., Kornelson, K., Larson, D., and Weber, E. (2007). Frames for undergraduates . American Mathematical Society, Providence, RI.
  119. Han, D. and Larson, D. R. (2000). Frames, bases and group representations. Mem. Amer. Math. Soc. , 147(697):x+94.
  120. Heil, C. (2011). A basis theory primer . Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York.
  121. Heil, C. (2013). What is …\ldots a frame? Notices Amer. Math. Soc. , 60(6):748--750.
  122. Hilding, S. H. (1948). Note on completeness theorems of P aley- W iener type. Ann. of Math. (2) , 49:953--955.
  123. Holub, J. R. (1994). Pre-frame operators, B esselian frames, and near- R iesz bases in H ilbert spaces. Proc. Amer. Math. Soc. , 122(3):779--785.
  124. James, R. C. (1982). Bases in B anach spaces. Amer. Math. Monthly , 89(9):625--640.
  125. Jamioikowski, A. (2010). An introduction to frames and their applications to quantum optics. In Quantum bio-informatics III , volume 26 of QP--PQ: Quantum Probab. White Noise Anal. , pages 147--154. World Sci. Publ., Hackensack, NJ.
  126. Janssen, A. J. E. M. (1996). Some W eyl- H eisenberg frame bound calculations. Indag. Math. (N.S.) , 7(2):165--183.
  127. Janssen, A. J. E. M. (2003). On generating tight G abor frames at critical density. J. Fourier Anal. Appl. , 9(2):175--214.
  128. Janssen, A. J. E. M. and Strohmer, T. (2002). Hyperbolic secants yield G abor frames. Appl. Comput. Harmon. Anal. , 12(2):259--267.
  129. Jiménez-Vargas, A., Sepulcre, J. M., and Villegas-Vallecillos, M. (2014). Lipschitz compact operators. J. Math. Anal. Appl. , 415(2):889--901.
  130. Johnson, W. B., Rosenthal, H. P., and Zippin, M. (1971). On bases, finite dimensional decompositions and weaker structures in B anach spaces. Israel J. Math. , 9:488--506.
  131. Kadec, M. I. (1964). The exact value of the P aley- W iener constant. Dokl. Akad. Nauk SSSR , 155:1253--1254.
  132. Kadison, R. V., Liu, Z., and Thom, A. (2020). A note on commutators in algebras of unbounded operators. Expo. Math. , 38(2):232--239.
  133. Kadison, R. V. and Ringrose, J. R. (1997). Fundamentals of the theory of operator algebras. V ol. I : E lementary theory , volume 15. American Mathematical Society, Providence, RI.
  134. Kaftal, V., Larson, D. R., and Zhang, S. (2009). Operator-valued frames. Trans. Amer. Math. Soc. , 361(12):6349--6385.
  135. Kaftal, V., Ng, P. W., and Zhang, S. (2014). Commutators and linear spans of projections in certain finite C∗C^* -algebras. J. Funct. Anal. , 266(4):1883--1912.
  136. Kalton, N. J. (2004). Spaces of L ipschitz and H ölder functions and their applications. Collect. Math. , 55(2):171--217.
  137. Kalton, N. J. and Lancien, G. (2008). Best constants for L ipschitz embeddings of metric spaces into c0c_0 . Fund. Math. , 199(3):249--272.
  138. Kashin, B. S. and Kulikova, T. Y. (2002). A remark on the description of frames of general form. Mat. Zametki , 72(6):941--945.
  139. Koehler, D. O. (1971). A note on some operator theory in certain semi-inner-product spaces. Proc. Amer. Math. Soc. , 30:363--366.
  140. Kovacevic, J. and Chebira, A. (2007). Life beyond bases: The advent of frames (part I ). IEEE Signal Processing Magazine , 24(4):86--104.
  141. Laustsen, N. J. (2002). Commutators of operators on B anach spaces. J. Operator Theory , 48(3, suppl.):503--514.
  142. Leon, S. J., Bjorck, A. k., and Gander, W. (2013). Gram- S chmidt orthogonalization: 100 years and more. Numer. Linear Algebra Appl. , 20(3):492--532.
  143. Levinson, N. (1936). On non-harmonic F ourier series. Ann. of Math. (2) , 37(4):919--936.
  144. Levy, E. and Shalit, O. M. (2014). Dilation theory in finite dimensions: the possible, the impossible and the unknown. Rocky Mountain J. Math. , 44(1):203--221.
  145. Li, D. F. and Sun, W. C. (2009). Expansion of frames to tight frames. Acta Math. Sin. (Engl. Ser.) , 25(2):287--292.
  146. Li, S. (1995). On general frame decompositions. Numer. Funct. Anal. Optim. , 16(9-10):1181--1191.
  147. Li, S. and Ogawa, H. (2004). Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. , 10(4):409--431.
  148. Limaye, B. V. (2014). Functional analysis . New Age International Publishers Limited, New Delhi.
  149. Liu, R. and Ruan, Z.-J. (2016). Cb-frames for operator spaces. J. Funct. Anal. , 270(11):4280--4296.
  150. Lu, T.-T. and Shiou, S.-H. (2002). Inverses of 2×22\times 2 block matrices. Comput. Math. Appl. , 43(1-2):119--129.
  151. Lumer, G. (1961). Semi-inner-product spaces. Trans. Amer. Math. Soc. , 100:29--43.
  152. Mallat, S. (2009). A wavelet tour of signal processing: The sparse way . Elsevier/Academic Press, Amsterdam, third edition.
  153. Marcoux, L. W. (2006). Sums of small number of commutators. J. Operator Theory , 56(1):111--142.
  154. Marcoux, L. W. (2010). Projections, commutators and L ie ideals in C∗C^* -algebras. Math. Proc. R. Ir. Acad. , 110A(1):31--55.
  155. Murphy, G. J. (1990). C∗C^* -algebras and operator theory . Academic Press, Inc., Boston, MA.
  156. Okoudjou, K. A., editor (2016). Finite frame theory: A complete introduction to overcompleteness , volume 73 of Proceedings of Symposia in Applied Mathematics . AMS Short Course Lecture Notes, American Mathematical Society, Providence, RI.
  157. Paley, R. E. A. C. and Wiener, N. (1987). Fourier transforms in the complex domain , volume 19 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI.
  158. Parrott, S. (1970). Unitary dilations for commuting contractions. Pacific J. Math. , 34:481--490.
  159. Paulsen, V. (2002). Completely bounded maps and operator algebras , volume 78 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge.
  160. Pedersen, G. K. (2018). C∗C^* -algebras and their automorphism groups . Pure and Applied Mathematics (Amsterdam). Academic Press, London.
  161. Pelczynski, A. (1971). Any separable B anach space with the bounded approximation property is a complemented subspace of a B anach space with a basis. Studia Math. , 40:239--243.
  162. Peng, I. and Waldron, S. (2002). Signed frames and H adamard products of G ram matrices. Linear Algebra Appl. , 347:131--157.
  163. Pesenson, I., Le Gia, Q. T., Mayeli, A., Mhaskar, H., and Zhou, D.-X., editors (2017). Frames and other bases in abstract and function spaces: Novel methods in harmonic analysis. Vol. 1 . Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham.
  164. Pietsch, A. (2007). History of B anach spaces and linear operators . Birkhäuser Boston, Inc., Boston, MA.
  165. Pisier, G. (2001). Similarity problems and completely bounded maps , volume 1618 of Lecture Notes in Mathematics . Springer-Verlag, Berlin.
  166. Popa, S. (1982). On commutators in properly infinite W∗W^{\ast} -algebras. In Invariant subspaces and other topics ( T imişoara/ H erculane, 1981) , volume 6 of Operator Theory: Adv. Appl. , pages 195--207. Birkhäuser, Basel-Boston, Mass.
  167. Putnam, C. R. (1967). Commutation properties of H ilbert space operators and related topics . Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36. Springer-Verlag New York, Inc., New York.
  168. Rahimi, A. and Balazs, P. (2010). Multipliers for pp - B essel sequences in B anach spaces. Integral Equations Operator Theory , 68(2):193--205.
  169. Rainone, T. (2007). A survey of J . von N eumann's inequality, M aster's T hesis, M c G ill U niversity, M ontreal, Q uebec, C anada.
  170. Rauhut, H. and Ullrich, T. (2011). Generalized coorbit space theory and inhomogeneous function spaces of B esov- L izorkin- T riebel type. J. Funct. Anal. , 260(11):3299--3362.
  171. Sarason, D. (1965). On spectral sets having connected complement. Acta Sci. Math. (Szeged) , 26:289--299.
  172. Schaffer, J. J. (1955). On unitary dilations of contractions. Proc. Amer. Math. Soc. , 6:322.
  173. Schatten, R. (1960). Norm ideals of completely continuous operators . Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 27. Springer-Verlag, Berlin-Göttingen-Heidelberg.
  174. Schneeberger, C. (1971). Commutators on a separable LpL^{p} -space. Proc. Amer. Math. Soc. , 28:464--472.
  175. Shalit, O. M. (2021). Dilation theory: a guided tour. In Operator theory, functional analysis and applications , volume 282 of Oper. Theory Adv. Appl. , pages 551--623. Birkhäuser/Springer.
  176. Shor, P. W. (2004). The adaptive classical capacity of a quantum channel, or information capacities of three symmetric pure states in three dimensions. IBM Journal of Research and Development , 48(1):115--137.
  177. Simon, B. (2015). Harmonic analysis . A Comprehensive Course in Analysis, Part 3. American Mathematical Society, Providence, RI.
  178. Skrettingland, E. (2020). Quantum harmonic analysis on lattices and G abor multipliers. J. Fourier Anal. Appl. , 26(3):48.
  179. Sondergaard, P. L. (2007). F inite D iscrete G abor A nalysis, P h D T hesis, T echnical U niversity of D enmark.
  180. Stasinski, A. (2016). Similarity and commutators of matrices over principal ideal rings. Trans. Amer. Math. Soc. , 368(4):2333--2354.
  181. Stoeva, D. T. (2009). XdX_d -frames in B anach spaces and their duals. Int. J. Pure Appl. Math. , 52(1):1--14.
  182. Stoeva, D. T. (2012). Perturbation of frames in B anach spaces. Asian-Eur. J. Math. , 5(1):1250011, 15.
  183. Stoeva, D. T. (2020). On a characterization of R iesz bases via biorthogonal sequences. J. Fourier Anal. Appl. , 26(4):Paper No. 67, 5.
  184. Stoeva, D. T. and Balazs, P. (2020). A survey on the unconditional convergence and the invertibility of frame multipliers with implementation. In Sampling: Theory and Applications: A Centennial Celebration of Claude Shannon , Appl. Numer. Harmon. Anal., pages 169--192. Birkhäuser/Springer, New York.
  185. Stroescu, E. (1973). Isometric dilations of contractions on B anach spaces. Pacific J. Math. , 47:257--262.
  186. Strohmer, T. (2001). Approximation of dual G abor frames, window decay, and wireless communications. Appl. Comput. Harmon. Anal. , 11(2):243--262.
  187. Strohmer, T. and Heath, Jr., R. W. (2003). Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. , 14(3):257--275.
  188. Sun, W. (2006). GG -frames and gg - R iesz bases. J. Math. Anal. Appl. , 322(1):437--452.
  189. Sun, W. (2007). Stability of gg -frames. J. Math. Anal. Appl. , 326(2):858--868.
  190. Sz.-Nagy, B. (1953). Sur les contractions de lespace de H ilbert. Acta Sci. Math. (Szeged) , 15:87--92.
  191. Sz.-Nagy, B. and Foias, C. (1971). The ``lifting theorem'' for intertwining operators and some new applications. Indiana Univ. Math. J. , 20(10):901--904.
  192. Sz.-Nagy, B., Foias, C., Bercovici, H., and Kérchy, L. (2010). Harmonic analysis of operators on H ilbert space . Universitext. Springer, New York.
  193. Takesaki, M. (2002). Theory of operator algebras. I , volume 124 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin.
  194. Tao, T. (2019). Commutators close to the identity. J. Operator Theory , 82(2):369--382.
  195. Terekhin, P. A. (2004). Representation systems and projections of bases. Mat. Zametki , 75(6):944--947.
  196. Terekhin, P. A. (2009). Banach frames in the affine synthesis problem. Mat. Sb. , 200(9):127--146.
  197. Terekhin, P. A. (2010). Frames in a B anach space. Funktsional. Anal. i Prilozhen. , 44(3):50--62.
  198. Thomas, S. M. (2012). Approximate S chauder frames for Rn\mathbb{R}^n, Masters Thesis, St. Louis University, St. Louis, MO .
  199. Valentine, F. A. (1945). A L ipschitz condition preserving extension for a vector function. Amer. J. Math. , 67:83--93.
  200. van Eijndhoven, S. J. L. (1996). Hilding's theorem for banach spaces. RANA: reports on applied and numerical analysis, volume 9612: Technische Universiteit Eindhoven , pages 1--6.
  201. Varopoulos, N. T. (1974). On an inequality of von N eumann and an application of the metric theory of tensor products to operators theory. J. Functional Analysis , 16:83--100.
  202. von Neumann, J. (1930). Allgemeine E igenwerttheorie H ermitescher F unktionaloperatoren. Math. Ann. , 102(1):49--131.
  203. von Neumann, J. (1951). Eine S pektraltheorie für allgemeine O peratoren eines unitären R aumes. Math. Nachr. , 4:258--281.
  204. Waldron, S. F. D. (2018). An introduction to finite tight frames . Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York.
  205. Weaver, N. (2018). Lipschitz algebras . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
  206. Wielandt, H. (1949). Über die U nbeschränktheit der O peratoren der Q uantenmechanik. Math. Ann. , 121:21.
  207. Wintner, A. (1947). The unboundedness of quantum-mechanical matrices. Phys. Rev. (2) , 71:738--739.
  208. Wold, H. (1954). A study in the analysis of stationary time series . Almqvist & Wiksell, Stockholm.
  209. Yood, B. (2008). Commutators in B anach ∗*-algebras. Studia Math. , 186(1):1--10.
  210. Young, R. M. (1980). An introduction to nonharmonic F ourier series . Academic Press, Inc., New York.
  211. Zhu, K. H. (1993). An introduction to operator algebras . Studies in Advanced Mathematics. CRC Press, Boca Raton, FL.

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  1. v1Submitted by K. MAHESH KRISHNAInitial depositCurrentOct 28, 2026