To a generalized tight continuous frame in a Hilbert space $\mathcal{H}$, indexed by a locally compact space $\Sigma$ endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on $\Sigma$ in spaces of vectors comparable with $\mathcal{H}$. If the continuous frame is provided by the action of a suitable family of bounded operators on a fixed window, a symbolic calculus emerges, assigning operators in $\mathcal{H}$ to functions on $\Sigma$. We give some criteria of relative compactness for sets and for families of compact operators, involving tightness properties in terms of objects canonically associated to the frame. Particular attention is dedicated to a magnetic version of the pseudodifferential calculus.
References
Ph. M. Anselone, Compactness properties of sets of operators and their adjoints, Math. Z. 113 (1970), 233–236. MR261397 10.1007/BF01110195 Ph. M. Anselone, Compactness properties of sets of operators and their adjoints, Math. Z. 113 (1970), 233–236. MR261397 10.1007/BF01110195
I. Beltiţă and D. Beltiţă, Magnetic Pseudo-differential Weyl calculus on nilpotent Lie groups. Ann. Global Anal. Geom. 36 (2009), no. 3, 293–322. MR2544305 10.1007/s10455-009-9166-8 I. Beltiţă and D. Beltiţă, Magnetic Pseudo-differential Weyl calculus on nilpotent Lie groups. Ann. Global Anal. Geom. 36 (2009), no. 3, 293–322. MR2544305 10.1007/s10455-009-9166-8
D. Beltiţă and I. Beltiţă, Continuity of magnetic Weyl calculs, J. Funct. Anal. 260 (2011), no. 7,. 1944-1968. MR2756145 10.1016/j.jfa.2011.01.004 D. Beltiţă and I. Beltiţă, Continuity of magnetic Weyl calculs, J. Funct. Anal. 260 (2011), no. 7,. 1944-1968. MR2756145 10.1016/j.jfa.2011.01.004
M. Dörfler, H. Feichtinger and K. Gröchenig, Compactness criteria in function spaces, Colloq. Math. 94 (2002), no.. 1, 37–50. MR1930200 10.4064/cm94-1-3 M. Dörfler, H. Feichtinger and K. Gröchenig, Compactness criteria in function spaces, Colloq. Math. 94 (2002), no.. 1, 37–50. MR1930200 10.4064/cm94-1-3
H.G. Feichtinger, Compactness in translational invariant Banach spaces of distributions and compact multipliers, J. Math. Anal. Appl. 102 (1984), no. 2, 289–327. MR755964 10.1016/0022-247X(84)90173-2 H.G. Feichtinger, Compactness in translational invariant Banach spaces of distributions and compact multipliers, J. Math. Anal. Appl. 102 (1984), no. 2, 289–327. MR755964 10.1016/0022-247X(84)90173-2
H.G. Feichtinger and K. Gröchenig, Banach spaces associated to integrable group representations and their atomic decompositions I, J. Funct. Anal. 86 (1989), 307–340. MR1021139 10.1016/0022-1236(89)90055-4 H.G. Feichtinger and K. Gröchenig, Banach spaces associated to integrable group representations and their atomic decompositions I, J. Funct. Anal. 86 (1989), 307–340. MR1021139 10.1016/0022-1236(89)90055-4
M. Fornasier and H. Rauhut, Continuous frames, function spaces and the discretization problem, J. Fourier Anal. Appl. 11 (2005), no. 3, 245–287. MR2167169 10.1007/s00041-005-4053-6 M. Fornasier and H. Rauhut, Continuous frames, function spaces and the discretization problem, J. Fourier Anal. Appl. 11 (2005), no. 3, 245–287. MR2167169 10.1007/s00041-005-4053-6
F. Galaz-Fontes, Note on compact sets of compact operators on a reflexive and separable space, Proc. Amer. Math. Soc. 126 (1998), no. 2, 587–588. MR1443386 10.1090/S0002-9939-98-04285-3 F. Galaz-Fontes, Note on compact sets of compact operators on a reflexive and separable space, Proc. Amer. Math. Soc. 126 (1998), no. 2, 587–588. MR1443386 10.1090/S0002-9939-98-04285-3
V. Georgescu and A. Iftimovici, Riesz-Kolmogorov compacity criterion, Ruelle theorem and Lorentz convergence on lcally compact Abelian groups, Potential Anal. 20 (2004), no. 3, 265–284. MR2032498 10.1023/B:POTA.0000010667.05599.56 V. Georgescu and A. Iftimovici, Riesz-Kolmogorov compacity criterion, Ruelle theorem and Lorentz convergence on lcally compact Abelian groups, Potential Anal. 20 (2004), no. 3, 265–284. MR2032498 10.1023/B:POTA.0000010667.05599.56
K. Gröchenig and T. Sthromer, Pseudodifferential operators on locally compact Abelian groups and Sjöstrand's symbol class, J. Reine Angew. Math. 613 (2007), 121–146. MR2377132 K. Gröchenig and T. Sthromer, Pseudodifferential operators on locally compact Abelian groups and Sjöstrand's symbol class, J. Reine Angew. Math. 613 (2007), 121–146. MR2377132
H. Hanche Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Exposition. Math. 28 (2010), no. 4, 385–394. MR2734454 10.1016/j.exmath.2010.03.001 H. Hanche Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Exposition. Math. 28 (2010), no. 4, 385–394. MR2734454 10.1016/j.exmath.2010.03.001
V. Iftimie, M. Măntoiu and R. Purice, Magnetic pseudodifferential operators, Publ. RIMS. 43 (2007), 585–623. MR2361789 10.2977/prims/1201012035 V. Iftimie, M. Măntoiu and R. Purice, Magnetic pseudodifferential operators, Publ. RIMS. 43 (2007), 585–623. MR2361789 10.2977/prims/1201012035
M. Măntoiu and R. Purice, The magnetic Weyl calculus, J. Math. Phys. 45 (2004), no. 4, 1394–1417. MR2043834 10.1063/1.1668334 M. Măntoiu and R. Purice, The magnetic Weyl calculus, J. Math. Phys. 45 (2004), no. 4, 1394–1417. MR2043834 10.1063/1.1668334
F. Mayoral, Compact sets of compact operators in absence of $l^1$, Proc. Amer. Math. Soc. 129 (2000), no. 1, 79–82. MR1784015 10.1090/S0002-9939-00-06007-X F. Mayoral, Compact sets of compact operators in absence of $l^1$, Proc. Amer. Math. Soc. 129 (2000), no. 1, 79–82. MR1784015 10.1090/S0002-9939-00-06007-X
T.W. Palmer, Totally bounded sets of precompact linear operators, Proc. Amer. Math. Soc. 20 (1969), 101–106. MR235425 10.1090/S0002-9939-1969-0235425-3 T.W. Palmer, Totally bounded sets of precompact linear operators, Proc. Amer. Math. Soc. 20 (1969), 101–106. MR235425 10.1090/S0002-9939-1969-0235425-3
N.V. Pedersen, Matrix coefficients and a Weyl correspondence for nilpotent Lie groups, Invent. Math. 118 (1994),. 1–36. MR1288465 10.1007/BF01231524 N.V. Pedersen, Matrix coefficients and a Weyl correspondence for nilpotent Lie groups, Invent. Math. 118 (1994),. 1–36. MR1288465 10.1007/BF01231524
H. Rauhut and T. Ullrich, Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type, J.. Funct. Anal. 260 (2011), no. 11, 3299–3362. MR2776571 10.1016/j.jfa.2010.12.006 H. Rauhut and T. Ullrich, Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type, J.. Funct. Anal. 260 (2011), no. 11, 3299–3362. MR2776571 10.1016/j.jfa.2010.12.006
E. Serrano, C. Pineiero and J.M. Delgado, Equicompact sets of operators defined on Banach spaces, Proc. Amer. Math. Soc. 134 (2005), no. 3, 689–695. MR2180885 10.1090/S0002-9939-05-08338-3 E. Serrano, C. Pineiero and J.M. Delgado, Equicompact sets of operators defined on Banach spaces, Proc. Amer. Math. Soc. 134 (2005), no. 3, 689–695. MR2180885 10.1090/S0002-9939-05-08338-3
