Decomposability of Weighted Composition Operators on $L^p$ of Atomic Measure Space

Shailesh Trivedi, Harish Chandra

Abstract


In this paper, we discuss the decomposability of weighted composition operator $uC_\phi$ on $L^p(X)(1\leq p<\infty)$ of a $\sigma$-finite atomic measure space $(X,\mathcal{S},\mu)$ with the assumption that $u\in L^\infty(X)$ and $|u|$ has positive ess inf. We prove that if the analytic core of $uC_\phi$ is zero and $uC_\phi$ is not quasinilpotent, then it is not decomposable. We also show that if $\phi$ is either injective almost everywhere or surjective almost everywhere but not both, then $uC_\phi$ is not decomposable. Finally, we give a necessary condition for decomposability of $uC_\phi$.

Full Text:

PDF

References


bibitem{Nordgren}

E. A. Nordgen, Composition Operators, Canad. J. Math., 20 (1968), 442--449.

bibitem{Foias}

I. Colojoara, C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968.

bibitem{Lange}

I. Erdelyi, R. Lange, Spectral Decompositions on Banach Spaces, Springer-Verlag, Berlin, 1977.

bibitem{Rho}

J. C. Rho, J. K. Yoo, (E)-Super-Decomposable Operators, J. Korean Math. Soc. 30 (1993), 211--227.

bibitem{Shapiro}

J. H. Shapiro, Decomposability and the cyclic behavior of parabolic composition operators, North-Holland Mathematics Studies 189 (2001), 143--157.

bibitem{Finch}

J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61--69.

bibitem{Laursen}

K. B. Laursen, M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000.

bibitem{Mbekhta}

M. Mbekhta, Sur la th'{e}orie spectrale locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), 621--631.

bibitem{Dunford1}

N. Dunford, Spectral theory. II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559--614.

bibitem{Dunford2}

N. Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321--354.

bibitem{Aiena}

P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, New York, 2004.

bibitem{AienaB}

P. Aiena, T. Biondi, Ascent, Descent, Quasi-nilpotent part and Analytic core of operators, Matematicki Vesnik 54 (2002), 57--70.

bibitem{AienaM}

P. Aiena, T. L. Miller, M. M. Neumann, On a Localised Single Valued Extension Property, Mathematical Proceedings of the Royal Irish Academy, 104A (1) (2004), 17--34.

bibitem{Kumar}

P. Kumar, A Study of Composition Operators on $l^p$ spaces, Thesis, Banaras Hindu University, Varanasi(2011).

bibitem{Vrbova}

P. Vrbov'{a}, On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23(98) (1973), 483--492.

bibitem{Smith}

R. C. Smith, Local spectral theory for invertible composition operators on $H^p$, Integr. Equat. Oper. Th. 25 (1996), 329--335.

bibitem{Harte}

R. Harte, On Local Spectral Theory II, Functional Analysis, Approximation and Computation

:1 (2010), 67--71.

bibitem{Manhas}

R. K. Singh, J. S. Manhas, Composition Operators on Function Spaces, North-Holland, New York, 1993.

bibitem{Singh}

R. K. Singh, T. Veluchamy, Atomic measure spaces and essentially normal composition operators, Bulletin of the Australian Mathematical Society, 27 (1983), 259--267.

bibitem{Trivedi}

S. Trivedi, H. Chandra, Some results on local spectral theory of Composition operators on $l^p$ spaces, textit{to appear in Matematicki Vesnik, $mv.mi.sanu.ac.rs/Papers/MV2013_003.pdf$}

bibitem{Miller}

T. L. Miller, V. G. Miller, M. M. Neumann, Local spectral properties of weighted shifts. J. Operator Theory 1 (2004), 71--88.


Refbacks

  • There are currently no refbacks.