2026/07/03 by M. Abtahi, Mortaza Abtahi, Ali Zamani +1
Mathematics · #Advanced Banach Space Theory #Holomorphic and Operator Theory #Approximation Theory and Sequence Spaces
paper · doi:10.1080/03081087.2026.2651902
Let B(H) represent the space of bounded operators on a Hilbert space H. Our primary objective is to extend the concept of the joint numerical range from finite systems, of the form (T1,…,Tn) in B(H)n, to infinite systems, spanning a broad spectrum that includes infinite sequences (T1,T2,…) of operators on one end, and continuous functions f:Ω→B(H), where Ω is a compact Hausdorff space, on the other. To this end, we formulate the system as a linear operator T:E∗→B(H), where E is a suitable Banach space. If T is weak∗ continuous on bounded sets, the joint numerical range W(T) of T is a totally bounded subset of E. We also study the joint algebraic numerical range V(T), and show that it equals the closed convex hull of W(T). Notably, any n-tuple (T1,…,Tn) is formulated as a linear operator T:Cn→B(H), given by T(β1,…,βn)=β1T1+⋯+βnTn, and the joint numerical range of T coincides with that of the n-tuple.