Absolutely Summing Operators from the Disc Algebra

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

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Norm estimates of certain p-absolutely summing operators

Functional Analysis and Its Applications ◽

10.1007/bf01076251 ◽

1978 ◽

Vol 12 (2) ◽

pp. 94-101 ◽

Cited By ~ 2

Author(s):

E. D. Gluskin

Keyword(s):

Norm Estimates ◽

Absolutely Summing Operators

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HOMOMORPHISMS OF BANACH ALGEBRAS WITH RANGE IN C1[0, 1]

International Journal of Mathematics ◽

10.1142/s0129167x94000115 ◽

1994 ◽

Vol 05 (02) ◽

pp. 201-212 ◽

Cited By ~ 1

Author(s):

HERBERT KAMOWITZ ◽

STEPHEN SCHEINBERG

Keyword(s):

Analytic Functions ◽

Compact Hausdorff Space ◽

Unit Disc ◽

Hausdorff Space ◽

Banach Algebras ◽

Locally Compact ◽

Disc Algebra ◽

Uniform Algebras ◽

Closed Subalgebra ◽

Uniformly Closed

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

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