Complex symmetric composition operators on weighted Hardy spaces

Abstract

Let φ \varphi be an analytic self-map of the open unit disk D \mathbb {D} . We study the complex symmetry of composition operators C φ C_\varphi on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D \mathbb {D} that is not an elliptic automorphism, we establish that if C φ C_{\varphi } is complex symmetric, then either φ ( 0 ) = 0 \varphi (0)=0 or φ \varphi is linear. In the case of weighted Bergman spaces A α 2 A^{2}_{\alpha } , we find the non-automorphic linear fractional symbols φ \varphi such that C φ C_{\varphi } is complex symmetric.

3-Complex Symmetric and Complex Normal Weighted Composition Operators on the Weighted Bergman Spaces of the Half-Plane

“…At the beginning, research was mainly focused on the study of complex symmetric operators on abstract Hilbert spaces (see [3][4][5]). As research continued, experts and scholars began to consider some special complex symmetric operators (such as composition operators and weighted composition operators) on analytic function spaces (see [6][7][8][9][10][11][12][13][14][15]).…”

Generalized weighted composition operators on weighted Hardy spaces

“…In recent decades, complex symmetric composition operators and weighted composition operators acting on some Hilbert spaces of analytic functions have been studied considerably. See [3][4][5][6][7][8][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]28] for more results on complex symmetric operators.…”

Composition operators on Hardy-Smirnov spaces

“…In [28, Theorem 2.5], Narayan, Sievewright and Tjani have shown that if Φ is an analytic self-map of U which is not an elliptic automorphism and C Φ is complex symmetric on H 2 (U), then Φ(0) = 0 or Φ is linear. They also characterized the non-automorphic linear self-maps of U that induce complex symmetric composition operators (see [28,Theorem 3.1]). In [29], Noor and Severiano characterized all linear fractional self-maps Φ of C + that induce complex symmetric composition operators C Φ on H 2 (C + ).…”