## Essentially hyponormal weighted composition operators on the Hardy and weighted Bergman spaces.(English)Zbl 07104710

Summary: Let $$\varphi$$ be an analytic self-map of the open unit disk $$\mathbb{D}$$ and let $$\psi$$ be an analytic function on $$\mathbb{D}$$ such that the weighted composition operator $$C_{\psi,\varphi}$$ defined by $$C_{\psi,\varphi}(f)=\psi f\circ\varphi$$ is bounded on the Hardy and weighted Bergman spaces. We characterize those weighted composition operators $$C_{\psi,\varphi}$$ on $$H^2$$ and $$A_{\alpha}^2$$ that are essentially hypo-normal, when $$\varphi$$ is a linear-fractional non-automorphism.

### MSC:

 47B33 Linear composition operators 47B20 Subnormal operators, hyponormal operators, etc.
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### References:

 [1] P.S. Bourdon, Spectra of some composition operators and associated weighted composition operators, J. Oper. Theory 67 (2012), 537-560. · Zbl 1262.47036 [2] P.S. Bourdon and S.K. Narayan, Normal weighted composition operators on the Hardy space $$H^2(U)$$, J. Math. Anal. Appl. 367 (2010), 278-286. · Zbl 1195.47013 [3] J.B. Conway, Functions of one complex variable, 2nd ed., Springer (1978). [4] J.B. Conway, A course in functional analysis, 2nd ed., Springer (1990). · Zbl 0706.46003 [5] J.B. Conway, The theory of subnormal operators, Amer. Math. Soc., Providence, RI (1991). · Zbl 0743.47012 [6] J.B. Conway, A course in operator theory, Graduate Studies in Math. 21, Amer. Math. Soc., Providence, RI (2000). [7] C.C. Cowen, Composition operators on $$H^2$$, J. Operator Theory 9 (1983), 77-106. · Zbl 0504.47032 [8] C.C. Cowen, Linear fractional composition operators on $$H^2$$, Integr. Equ. Oper. Theory 11 (1988), 151-160. · Zbl 0638.47027 [9] C.C. Cowen, S. Jung and E. Ko, Normal and cohyponormal weighted composition operators on $$H^2$$, pp. 69-85 in Operator theory in harmonic and non-commutative analysis (Sydney, 2012), Operator Theory: Advances and Applications 240, Springer (2014). [10] C.C. Cowen, E. Ko, D. Thompson and F. Tian, Spectra of some weighted composition operators on $$H^2$$, Acta Sci. Math. (Szeged) 82 (2016), 221-234. · Zbl 1399.47083 [11] C.C. Cowen and T.L. Kriete, Subnormality and composition operators on $$H^2$$, J. Funct. Anal. 81 (1988), 298-319. · Zbl 0669.47012 [12] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL (1995). · Zbl 0873.47017 [13] M. Fatehi and M. Haji Shaabani, Some essentially normal weighted composition operators on the weighted Bergman spaces, Complex Var. Elliptic Equ. 60 (2015), 1205-1216. · Zbl 1483.47047 [14] M. Fatehi, M. Haji Shaabani and D. Thompson, Quasinormal and hyponormal weighted composition operators on $$H^{ 2}$$ and $$A^2_{\alpha}$$ with linear fractional compositional symbol, Complex Anal. Oper. Theory 12 (2018), 1767-1778. [15] P. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), 503-513. · Zbl 0902.47030 [16] T.L. Kriete, B.D. MacCluer and J.L. Moorhouse, Toeplitz-composition $$C^{\ast}$$-algebras, J. Operator Theory 58 (2007), 135-156. · Zbl 1134.47303 [17] T. Le, Self-adjoint, unitary, and normal weighted composition operators in several variables, J. Math. Anal. Appl. 395 (2012), 596-607. · Zbl 1266.47043 [18] B.D. MacCluer, S.K. Narayan and R.J. Weir, Commutators of composition operators with adjoints of composition operators on weighted Bergman spaces, Complex Var. Elliptic Equ. 58 (2013), 35-54. · Zbl 1285.47031 [19] J.H. Shapiro, Composition operators and classical function theory, Springer (1993). · Zbl 0791.30033 [20] J.G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453-1458. · Zbl 0129.08701 [21] N. Zorboska, Hyponormal composition operators on the weighted Hardy spaces, Acta Sci. Math. (Szeged) 55 (1991), 399-402. · Zbl 0894.47023
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