## Rocky Mountain Journal of Mathematics

### Essentially hyponormal weighted composition operators on the Hardy and weighted Bergman spaces

Mahsa Fatehi

#### Abstract

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.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1129-1142.

Dates
First available in Project Euclid: 29 August 2019

https://projecteuclid.org/euclid.rmjm/1567044032

Digital Object Identifier
doi:10.1216/RMJ-2019-49-4-1129

Mathematical Reviews number (MathSciNet)
MR3998914

Zentralblatt MATH identifier
07104710

Subjects
Primary: 47B33: Composition operators
Secondary: 47B20: Subnormal operators, hyponormal operators, etc.

#### Citation

Fatehi, Mahsa. Essentially hyponormal weighted composition operators on the Hardy and weighted Bergman spaces. Rocky Mountain J. Math. 49 (2019), no. 4, 1129--1142. doi:10.1216/RMJ-2019-49-4-1129. https://projecteuclid.org/euclid.rmjm/1567044032

#### References

• P.S. Bourdon, Spectra of some composition operators and associated weighted composition operators, J. Oper. Theory 67 (2012), 537–560.
• 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.
• J.B. Conway, Functions of one complex variable, 2nd ed., Springer (1978).
• J.B. Conway, A course in functional analysis, 2nd ed., Springer (1990).
• J.B. Conway, The theory of subnormal operators, Amer. Math. Soc., Providence, RI (1991).
• J.B. Conway, A course in operator theory, Graduate Studies in Math. 21, Amer. Math. Soc., Providence, RI (2000).
• C.C. Cowen, Composition operators on $H^{2}$, J. Operator Theory 9 (1983), 77–106.
• C.C. Cowen, Linear fractional composition operators on $H^{2}$, Integr. Equ. Oper. Theory 11 (1988), 151–160.
• 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).
• 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.
• C.C. Cowen and T.L. Kriete, Subnormality and composition operators on $H^{2}$, J. Funct. Anal. 81 (1988), 298–319.
• C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL (1995).
• 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.
• 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.
• P. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), 503–513.
• T.L. Kriete, B.D. MacCluer and J.L. Moorhouse, Toeplitz-composition $C^{\ast}$-algebras, J. Operator Theory 58 (2007), 135–156.
• T. Le, Self-adjoint, unitary, and normal weighted composition operators in several variables, J. Math. Anal. Appl. 395 (2012), 596–607.
• 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.
• J.H. Shapiro, Composition operators and classical function theory, Springer (1993).
• J.G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453–1458.
• N. Zorboska, Hyponormal composition operators on the weighted Hardy spaces, Acta Sci. Math. (Szeged) 55 (1991), 399–402.