Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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BIISOMETRIC OPERATORS AND BIORTHOGONAL SEQUENCES

  • Kubrusly, Carlos (Applied Mathematics Department Federal University) ;
  • Levan, Nhan (Electrical Engineering Department University of California)
  • Received : 2018.03.19
  • Accepted : 2019.04.01
  • Published : 2019.05.31
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Abstract

It is shown that a pair of Hilbert space operators V and W such that $V^*W=I$ (called a biisometric pair) shares some common properties with unilateral shifts when orthonormal bases are replaced with biorthogonal sequences, and it is also shown how such a pair of biisometric operators yields a pair of biorthogonal sequences which are shifted by them. These are applied to a class of Laguerre operators on $L^2[0,{\infty})$.

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References

  1. C. S. Kubrusly, The Elements of Operator Theory, second edition, Birkhauser/Springer, New York, 2011.
  2. C. S. Kubrusly and N. Levan, Dual-shift decomposition of Hilbert space, in Semigroups of operators: theory and applications (Rio de Janeiro, 2001), 145-157, Optimization Software, New York, 2002.
  3. C. S. Kubrusly and N. Levan, Abstract wavelets generated by Hilbert space shift operators, Adv. Math. Sci. Appl. 16 (2006), no. 2, 643-660.
  4. N. Levan and C. S. Kubrusly, Dual-shift decomposition and wavelets, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), no. 2, 1450014, 15 pp.
  5. N. Levan and T. Nambu, On a class of shifts and shift semigroups of finite multiplicities, Japan J. Indust. Appl. Math. 13 (1996), no. 1, 93-105. https://doi.org/10.1007/BF03167300
  6. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Springer-Verlag, Berlin, 1977.
  7. P. M. Makila and J. R. Partington, Laguerre and Kautz shift approximations of delay systems, Internat. J. Control 72 (1999), no. 10, 932-946. https://doi.org/10.1080/002071799220678
  8. I. Singer, Bases in Banach Spaces. I, Springer-Verlag, New York, 1970.
  9. R. M. Young, On complete biorthogonal systems, Proc. Amer. Math. Soc. 83 (1981), no. 3, 537-540. https://doi.org/10.1090/S0002-9939-1981-0627686-9