Operator Theory on Noncommutative Polyballs

Gelu Popescu ()
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Gelu Popescu: The University of Texas at San Antonio, Department of Mathematics

Chapter 45 in Operator Theory, 2026, pp 1375-1451 from Springer

Abstract: Abstract The noncommutative regular polyball B n $${\mathbf {B}}_{\mathbf {n}}$$ , n = ( n 1 , … , n k ) ∈ ℕ k $$\mathbf {n}=(n_1,\ldots , n_k)\in {\mathbb N}^k$$ , is a noncommutative analogue of the scalar polyball ( ℂ n 1 ) 1 × ⋯ × ( ℂ n k ) 1 $$({\mathbb C}^{n_1})_1\times \cdots \times ({\mathbb C}^{n_k})_1$$ and has a universal model S : = { S i , j } $$ \mathbf {S}:=\{{\mathbf {S}}_{i,j}\}$$ consisting of the left creation operators acting on the tensor product F 2 ( H n 1 ) ⊗ ⋯ ⊗ F 2 ( H n k ) $$F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$$ of full Fock spacesFock space. If B ( ℋ ) $$B({\mathcal H})$$ denotes the algebra of all bounded linear operators on a Hilbert space ℋ $${\mathcal H}$$ , then B n ( ℋ ) − $${\mathbf {B}}_{\mathbf {n}}({\mathcal H})^-$$ coincides with the closure in the operator norm of the set of all elements X = { X i , j } ∈ B ( ℋ ) n 1 + ⋯ + n k $$\mathbf {X}=\{X_{i,j}\}\in B({\mathcal H})^{n_1+\cdots + n_k}$$ that admit S $$\mathbf {S}$$ as universal model, i.e., there is a Hilbert space D $${\mathcal D}$$ such that X i , j ∗ = ( S i , j ∗ ⊗ I D ) | ℋ $$X_{i,j}^*=({\mathbf {S}}_{i,j}^*\otimes I_{\mathcal D})|_{\mathcal H}$$ , where ℋ $${\mathcal H}$$ is a jointly coinvariant subspace under all S i , j ⊗ I D $${\mathbf {S}}_{i,j}\otimes I_{\mathcal D}$$ . This chapter is a survey on the multivariable operator theory on noncommutative polyballs B n $${\mathbf {B}}_{\mathbf {n}}$$ , their universal models S $$\mathbf {S}$$ , and the algebras they generate: the polyball algebra A n $$\boldsymbol {\mathcal A}_{\mathbf {n}}$$ , the Hardy algebra F n ∞ $${\mathbf {F}}^\infty _{\mathbf {n}}$$ , and the C ∗ $$C^*$$ -algebra C ∗ ( S ) $$C^*(\mathbf {S})$$ . This is accompanied by the theory of free holomorphic functions and free pluriharmonic functions on noncommutative polyballs that are related, via noncommutative Berezin transforms, to the study of the operator algebras generated by the universal model S $$\mathbf {S}$$ , as well as to the theory of functions in several complex variables.

Keywords: Noncommutative polyball; Berezin transform; Fock space; Creation operators; Hardy algebras; Numerical invariants; Automorphisms; Toeplitz operators; Hyperbolic geometry; Similarity problems; Primary: 47-02; 47A45; 46L40; 47A13; Secondary: 47A56; 47A20; 46L52; 47A15; 47B35 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_73

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DOI: 10.1007/978-3-032-16356-1_73

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