Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Abstract Let $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> be a positive finite Borel measure on [0, 1). Cesàro-type operators $$C_{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that $$C_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of $$C_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments $$(\mu _n)_{n\in {\mathbb {N}}_0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:msub> </mml:math> . The continuity, compactness and spectrum of $$C_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> acting on Fréchet and (LB) Korenblum type spaces are also considered .