Volume 11, Issue 1

We study the general properties, point spectrum and spectrum of a weighted composition operator W m , φ {W}_{m,\varphi } with elliptic symbol φ \varphi on the unit ball B N {{\mathbb{B}}}_{N} of C N {{\mathbb{C}}}^{N} , and general weight m ∈ Hol ( B N ) m\in {\rm{Hol}}\left({{\mathbb{B}}}_{N}) . We give a complete description of the spectra in the majority of cases, and we provide inclusions in general.

This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n × n n\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is the complete characterization of the set of all Toeplitz operators that commute with a given operator. Researchers showed that when a sum S = T e i m θ f + T e i l θ g S={T}_{{e}^{im\theta }f}+{T}_{{e}^{il\theta }g} , where f f and g g are radial functions, commutes with a sum T = T e i p θ r ( 2 M + 1 ) p + T e i s θ r ( 2 N + 1 ) s T={T}_{{e}^{ip\theta }{r}^{\left(2M+1)p}}+{T}_{{e}^{is\theta }{r}^{\left(2N+1)s}} , then S S must be of the form S = c T S=cT , where c c is a constant. In this article, we will replace r ( 2 M + 1 ) p {r}^{\left(2M+1)p} and r ( 2 N + 1 ) s {r}^{\left(2N+1)s} with r n {r}^{n} and r d {r}^{d} , where n n and d d are in N {\mathbb{N}} , and we will show that the same result holds.

This article studies the level set Crouzeix (LSC) conjecture, which is a weak version of Crouzeix’s conjecture that applies to finite compressions of the shift. Among other results, this article establishes the LSC conjecture for several classes of 3 × 3 3\times 3 , 4 × 4 4\times 4 , and 5 × 5 5\times 5 matrices associated with compressions of the shift via a geometric analysis of their numerical ranges. This study also establishes Crouzeix’s conjecture for several classes of nilpotent matrices whose studies are motivated by related compressions of shifts.