On the spectrum of the lacunary Cesaro operator

In this paper, we extend the classical Cesaro operator by introducing the concept of the lacunary Cesaro operator, which incorporates the idea of lacunary sequences. We examine the operator's matrix representation and its adjoint, establishing the boundedness of these operators on Banach spaces, including the spaces of sequences that converge to zero, absolutely summable sequences, and bounded sequences. Additionally, we determine the spectrum of the operator, showing that it adapts to the structure of the underlying lacunary sequence. Notably, the spectral radius remains equal to one for a wide range of lacunary sequences.