The commutant and strong irreducibility of the coordinate function operators

Xingfang Feng () and Yucheng Li ()
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Xingfang Feng: Hebei Normal University
Yucheng Li: Hebei Normal University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 3, 667-674

Abstract: Abstract Let $${{\mathbb {S}}}^{n}$$ S n denote the unit sphere in $${\mathbb C}^{n}$$ C n . Let $$H^{2}({{\mathbb {S}}}^{n})$$ H 2 ( S n ) be the standard Hardy space given by the closure in $$L^{2}({{\mathbb {S}}}^{n})$$ L 2 ( S n ) of the polynomials in the coordinate functions of $$z_{1},\cdots , z_{n}$$ z 1 , ⋯ , z n . For 2-tuple $$\{T_{1}, T_{2}\}$$ { T 1 , T 2 } acting on $$H^{2}({{\mathbb {S}}}^{2})$$ H 2 ( S 2 ) , we prove that $${{\mathcal {A}}}^{\prime }(T_{j}){\mathop {\simeq }\limits ^{u}} \bigoplus \limits _{1}^{\infty }{M_z}$$ A ′ ( T j ) ≃ u ⨁ 1 ∞ M z , moreover, $$K_{0}(\mathcal{A}^{\prime }(T_{jk}))\cong {{\mathbb {Z}}}\,(j=1,2;k=0,1,\cdots )$$ K 0 ( A ′ ( T jk ) ) ≅ Z ( j = 1 , 2 ; k = 0 , 1 , ⋯ ) . Furthermore, the commutant and strong irreducibility corresponding to the 2-tuple are discussed.

Keywords: Commutant; Cowen-Douglas operators; Weighted shift operators; Strong irreducibility; 46J40; 47A15; 32A36 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00284-z

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