Fractional resolvent family generated by normal operators

Chen-Yu Li; Chen-Yu Li

doi:10.3934/math.20231213

AIMS Mathematics

2023, Volume 8, Issue 10: 23815-23832. doi: 10.3934/math.20231213

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Fractional resolvent family generated by normal operators

Chen-Yu Li ^,

College of Computer Science, Chengdu University, Chengdu, China

Received: 17 May 2023 Revised: 22 June 2023 Accepted: 07 July 2023 Published: 04 August 2023
MSC : 35R11, 47B02, 47A10

The main focus of this paper is on the relationship between the spectrum of generators and the regularity of the fractional resolvent family. We will give a counter-example to show that the point-spectral mapping theorem is not valid for $ \{S_{\alpha}(t)\} $ if $ \alpha \neq 1 $; and we show that if $ \{S_{\alpha}(t)\} $ is stable, then we can determine the decay rate by $ \sigma(A) $ and some examples are given; we also prove that $ S_{\alpha}(t)x $ has a continuous derivative of order $ \alpha\beta > 0 $ if and only if $ x \in D(I-A)^{\beta} $. The main method we used here is the resolution of identity corresponding to a normal operator $ A $ and spectral measure integral.
- fractional resolvent family,
- normal operator,
- spectral mapping theorem, stable resolvent
Citation: Chen-Yu Li. Fractional resolvent family generated by normal operators[J]. AIMS Mathematics, 2023, 8(10): 23815-23832. doi: 10.3934/math.20231213

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Abstract

The main focus of this paper is on the relationship between the spectrum of generators and the regularity of the fractional resolvent family. We will give a counter-example to show that the point-spectral mapping theorem is not valid for $ \{S_{\alpha}(t)\} $ if $ \alpha \neq 1 $; and we show that if $ \{S_{\alpha}(t)\} $ is stable, then we can determine the decay rate by $ \sigma(A) $ and some examples are given; we also prove that $ S_{\alpha}(t)x $ has a continuous derivative of order $ \alpha\beta > 0 $ if and only if $ x \in D(I-A)^{\beta} $. The main method we used here is the resolution of identity corresponding to a normal operator $ A $ and spectral measure integral.

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