G-frames as special frames

G-frames are generalizations of ordinary frames for Hilbert spaces. In the present paper we study frames, and operators on a special separable Hilbert C*-module, B(H,K), where H and K are Hilbert spaces, and we prove that every g-frame for H is a frame for B(H,K) and vice versa. Also, we derive some relationships between g-Riesz bases for H and Riesz bases in B(H,K). Similar results for orthogonal bases will be discussed.

G-frames as special frames

G-frames are generalizations of ordinary frames for Hilbert spaces. In the present paper we study frames, and operators on a special separable Hilbert C*-module, B(H,K), where H and K are Hilbert spaces, and we prove that every g-frame for H is a frame for B(H,K) and vice versa. Also, we derive some relationships between g-Riesz bases for H and Riesz bases in B(H,K). Similar results for orthogonal bases will be discussed.

___

  • Bodmann, B.G., Kribs, D.W., Paulsen, V.I.: Decoherence-insensitive quantum communication by optimal C ∗ encoding. IEEE Trans. Inform. Theory 53, 4738-4749 (2007).
  • Christensen, O.: An Introduction to Frames and Riesz Bases. Boston-Basel-Berlin. Birkh¨ auser 2002.
  • D’Attellis, C.E., Fernfedez-Berdaguer, E.M.: Wavelet Theory and Harmonic Analysis in Applied Sciences. Boston - Basel - Berlin. Birkh¨ auser 1997.
  • Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271-1283 (1986). Duffin, R., Schaeffer, A.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341-366 (1952).
  • Frank, M., Larson, D.: Frames in Hilbert C ∗ -modules and C ∗ -algebra. J. Operator Theory 48, 273-314 (2002).
  • Gr¨ ochenig, K.: Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Boston. MA, Birkh¨ auser 200 Hemmat, A. A., Gabardo, J. P.: Properties of oblique dual frames in shift-invariant systems. J. Math. Anal. Appl. 356, 346-354 (2009).
  • Kaplansky, I.: Algebra of type I. Ann. Math. 56, 460-472 (1952).
  • Lance, E.: Hilbert C ∗ -modules - a Toolkit for Operator Algebraists. London Mathematical Society Lecture Note Series v. 210. Cambridge, England. Cambridge University Press 1995.
  • Murphy, G. J.: C ∗ -algebra and Operator Theory. London. Academic Press 1990.
  • Paschke, W.: Inner product modules over B ∗ -algebra. Trans. Amer. Math. Soc. 182, 443-468 (1973).
  • Rieffel, M.: Morita equivalence for C ∗ -algebra. J. Pure Applied Algebra 5, 51-96 (1974).
  • Rudin, W.: Functional Analysis. New York. McGraw-Hill Book Company 1973.
  • Sun, W.: G-frames and g-Riesz bases. J. Math. Anal. 322, 437-452 (2006).
  • Wegga-Olsen, N.: K-theory and C ∗ -algebra - a Friendly Approach. Oxford, England. Oxford University Press 1993. Young, R.: An Introduction to Nonharmonic Fourier Series. New York. Academic Press 1980.