S-operators related to a finite measure space

ABSTRACT

In this work we introduce and study a class of linear operators related to a finite measure space, for which its $L^2$-Hilbert space is separable. These linear operators represent a generalization of pseudo-differential operators on $S^1$, where $S^1$ is the unit circle with centre at the origin. We call these operators, S-operators or generalized pseudo-differential operators on S, where $(S,\mathcal{B},m)$ is the corresponding finite measure space for which the $L^2$-space is separable. We give some $L^2$-boundedness and compactness results, and we also study the Hilbert–Schmidt property in connection with this class of linear operators. In the end we give some examples of finite measure spaces and corresponding S-operators for which we can apply our results.

Keywords:

• Measure space
• measurable function
• compact operator
• Hilbert–Schmidt operator
• pseudo-differential operator
• Young's inequality
• Parseval's identity
• Plancherel formula

2010 MSC SUBJECT CLASSIFICATIONS:

• Primary: 47B38
• 42A16
• 47G30
• Secondary: 46E30
• 42C10

Acknowledgements

The author is grateful to the anonymous referee for valuable remarks and comments which led to several improvements of the original paper.

Disclosure statement

No potential conflict of interest was reported by the author.