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In this article, we exhibit a large class of Banach spaces whose open unit balls are bounded symmetric homogeneous domains. These Banach spaces, which we call J*-algebras, are linear spaces of operators mapping one Hilbert space into another and have a kind of Jordan tripte product structure. In particular, all Hilbert spaces and all B*-algebras are… (More)

We discuss the Earle-Hamilton fixed point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents… (More)

This article considers the extension of V. A. Markov's theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain… (More)

We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange poly-nomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the… (More)

Our object is to study domains which are the region of negative deenite-ness of an operator-valued Hermitian form deened on a space of operators and to investigate the biholomorphic linear fractional transformations between them. This is a uniied setting in which to consider operator balls, operator half-planes, strictly J-contractive operators, strictly… (More)

Let X and Y be real normed linear spaces and let φ : X → R be a non-negative function satisfying φ(x + y) ≤ φ(x) + y for all x, y ∈ X. We show that there exist optimal constants c m,k such that if P : X → Y is any polynomial satisfying P (x) ≤ φ(x) m for all x ∈ X, thenˆD k P (x) ≤ c m,k φ(x) m−k whenever x ∈ X and 0 ≤ k ≤ m. We obtain estimates for these… (More)

Let φ : (−∞, ∞) → (0, ∞) be a given continuous even function and let m be a positive integer. We show that, with some additional restrictions on φ, there exist decreasing sequences x where equality holds with alternating signs at the corresponding sequence of points (and also at ±∞ for Q). Moreover, for any polynomial p of degree at most m, a) if |p(x j)| ≤… (More)