SPECTRA OF LINEAR FRACTIONAL COMPOSITION OPERATORS ON THE GROWTH SPACE AND BLOCH SPACE OF THE UPPER HALF-PLANE

In this article, we provide a complete description of the spectra of linear fractional composition operators acting on the growth space and Bloch space over the upper half-plane. In addition, we also prove that the norm, essential norm, spectral radius and essential spectral radius of a composition operator acting on the growth space are all equal.

Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

Reliability Analysis of a Two-Unit System with Connecting and Disconnecting Effect

A Two-Unit system with connecting and disconnecting effect is studied in this paper. By the method of Functional analysis strong continuous semi-group, the paper analyzes the restriction of essential spectral growth bound of the system operator. The restriction of essential spectral growth bound of the system operator and the change of the essential spectral radius after perturbation is analyzed. The essential spectral radius of the system operator is also discussed before and after perturbation. The results show that under some conditions the dynamic solution of the system is exponential stability and tends to the steady solution of the system. At last, we analyze the reliability of the system.