Algebras of polynomials generated by linear operators - Abtahi, M.; Zaj, F. (Carpathian Mathematical Publications, 2024)
Let E be a Banach space and A be a commutative Banach algebra with identity. Let P(E, A) be the space of A-valued polynomials on E generated by bounded linear operators (an n-homogenous polynomial in P(E,A) is of the form P=_i=1^∞ T^n_i, where T_i:E→ A, 1≤ i <∞, are bounded linear operators and _i=1^∞ \|T_i\|^n < ∞). For a compact set K in E, we let P(K, A) be the closure in C(K,A) of the restrictions P|_K of polynomials P in P(E,A). It is proved that P(K, A) is an A-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product P_N(K)_ε A, where P_N(K) is the uniform algebra on K generated by nuclear scalar-valued polynomials. The character space of P(K, A) is then identified with K_N× M(A), where K_N is the nuclear polynomially convex hull of K in E, and M(A) is the character space of A.