Spectrum and essential spectrum of linear combinations of composition operators on the Hardy space H2

Subject Areas : Statistics

M. Haji Shaabani ¹ , M. Fatehi ² , M. Farzi Haromi ³

1 - Professor, Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.
2 - Professor, Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran.
3 - Ph.D. Student, Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.

Received: 2018-02-21 Accepted : 2019-01-27 Published : 2019-05-01

Keywords: فضای هاردی, عملگر ترکیبی, طیف, طیف اساسی,

Abstract :

Let -----. For an analytic self-map --- of --- , Let --- be the composition operator with composite map --- so that ----. Let --- be a bounded analytic function on --- . The weighted composition operator --- is defined by --- . Suppose that --- is the Hardy space, consisting of all analytic functions defined on --- , whose Maclaurin cofficients are square summable. .....

References:

[1] P. S. Bourdon, Spectra of some composition operators and associated weighted composition operators,J. OperatorTheory 67 (2012), 537-560.

 [2] P. S. Bourdon, Spectra of composition operators with symbols in S(2), J. Operator Theory 75 (2016), 21-48.

[3] P. S. Bourdon and S. Narayan, Normal weighted composition operators on the Hardy space , J. Math. Anal. App. 367 (2010), 278-286.

[4] J. G. Caughran and H. J. Schwartz, Spectra of compact composition operators, Proc. Amer. Math. Soc. 51 (1975), 127-130.

[5] J. B. Conway, Functions of One Complex Variable, Second Edition, Springer-Verlag, New York, 1978.

[6] J. B. Conway, A Course in Functional Analysis, Second Edition, Springer-Verlag, New York, 1990.

[7] C‎. C. ‎Cowen, Composition operators on , J. Operator Theory 9 (1983), 77-106.

[8] C‎. C. ‎Cowen and E. Ko, Hermitian weighted composition operators on weighted Hardy spaces, Trans. Amer. Math. Soc. 362 (2010), 5771-5801.

[9] C‎. C. ‎Cowen, E. Ko, D. Thompson and F. Tian, Spectra of some weighted composition operators on , Acta Sci. Math. (Szeged) 82 (2016), 221-234.

 

[10] C‎. C. ‎Cowen and B‎. D. ‎MacCluer‎, Composition Operators on Spaces of Analytic Functions‎. Staudies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.

[11] J. A. Deddens, Analytic Toeplitz and composition operators, Canad. J. Math. 24 (1972), 859-865.

[12] M. Fatehi and M. Haji Shaabani, Certain nontrivially essentially self-adjoint weighted composition operators on H2 and A2, Complex Variables and Elliptic Equations. 59 (2014), 1626-1635.

[13] M. Fatehi and M. Haji Shaabani, Some essentially normal weighted composition operators on the weighted Bergman spaces, Complex Variables and Elliptic Equations. 60 (2015), 1205-1216.

[14] M. Fatehi and M. Haji Shaabani, Normal, cohyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces, Journal of the Korean Mathematical Society. 54 (2017), 599–612.

[15] M. Fatehi and M. Haji Shaabani, Norms of hyponormal weighted composition operators on the Hardy and weighted Bergman Spaces, Operators and Matrices. 12 (2018), 997–1007.

 [16] M. Fatehi, M. Haji Shaabani and D. Thompson, Quasinormal and Hyponormal Weighted Composition Operators on H2 and A2 with Linear Fractional Compositional Symbol, Complex Anal. Oper. Theory, 12 (2018), 1767–1778.

 

[17] M. Fatehi, Complex symmetric weighted composition operators, Complex Variables and Elliptic Equations, to appear.

 [18] G. Gunatillake, Spectrum of a compact weighted composition operator, Proc. Amer. Math. Soc. 135 (2) (2007), 461-467.

[19] G. Gunatillake, Invertible weighted composition operators, J. Funct. Anal. 261 (2011), 831-860.

[20] O. Hyvarinen, M. Lindstrom, I. Nieminen, and E. Saukko, Spectra of weighted composition operators with automorphic symbols, J. Funct. Anal. 265 (2013), 1749-1777.

[21] H. Kamowitz, The spectra of composition ‎operators on , J. Funct. Anal.18 (1975),132-150.

[22] T. L. Kriete, B. D. MacCluer and J. L. Moorhouse, Toeplitz- composition -algebras, J. Operator Theory. 58 (2007), 135-156.

[23] R. Lim, L. H. Khoi, Complex symmetric weighted composition operators  J. Math. Anal. Appl. 464 (2018), 101-118.

[24] S. K. Narayan, D. Sievewright and D. Thompson, Complex symmetric composition operators on H2, J. Math. Anal. Appl. 443 (2016), 625-630.

[25] E. A. Nordgren, Composition ‎operators, Canad. J. Math. 20 (1968), 442-449.

 

[26] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.

 

[27] D. Thompson, Normaloid weighted composition operators on H2. Journal of Mathematical Analysis and Applications, 467 (2018), 1153-1162.