2025/10/29 by Keshari, Dinesh Kumar, Pal, Avijit, Paul, Bhaskar
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2510.25666
A 7-tuple of commuting bounded operators T = (T1, …, T7) on a Hilbert space H is called a \textitΓE(3; 3; 1, 1, 1) -contraction if ΓE(3; 3; 1, 1, 1) is a spectral set for T. Let (S1, S2, S3) and (S1, S2) be tuples of commuting bounded operators defined on a Hilbert space H with SiSj = SjSi for 1 \leqslant i \leqslant 3 and 1 \leqslant j \leqslant 2. We say that S = (S1, S2, S3, S1, S2) is a ΓE(3; 2; 1, 2) -contraction if ΓE(3; 2; 1, 2) is a spectral set for S. We derive various properties of ΓE(3; 3; 1, 1, 1)-contractions and ΓE(3; 2; 1, 2)-contractions and establish a relationship between them. We discuss the fundamental equations for ΓE(3; 3; 1, 1,1 )-contractions and ΓE(3; 2; 1, 2)-contractions. We explore the structure of ΓE(3; 3; 1, 1, 1)-unitaries and ΓE(3; 2; 1, 2)-unitaries and elaborate on the relationship between them. We also study various properties of ΓE(3; 3; 1, 1, 1)-isometries and ΓE(3; 2; 1, 2)-isometries. We discuss the Wold Decomposition for a ΓE(3; 3; 1, 1, 1)-isometry and a ΓE(3; 2; 1, 2)-isometry. We further outline the structure theorem for a pure ΓE(3; 3; 1, 1, 1)-isometry and a pure ΓE(3; 2; 1, 2)-isometry.