Operator Spectral Similarity Of A Kind Of Fractals And Its’ Properties

Analysis in Hilbert spaces and spectral theory are an indispensable tool in modern mathematics, physics and Engineering Science. Many problems in mathematical physics, quantum mechanics, Signal processing and ergodic theory can be transformed into spectral problems. Spectrum is an important object of study in fractal analysis. By using analytic approach, this paper gives a criterion of operator spectral similarity and discusses the necessary conditions that graph Laplace operator△G, Markov operator△M, and adjacent matrix operator△A in a locally finite graph respectively spectrally similar to Laplace operator△G0in graph Go where graph G is a graph symmetric with respect to V(Go), the vertices of Go, and G0is a complete finite graph with V(G0)∈V(G).