K-bounded polynomials

DANIEL CARANDO; VERÓNICA DIMANT; BETINA DUARTE; SILVIA LASSALLE

PROCEEDINGS OF THE ROYAL IRISH ACADEMY SECTION A-MATHEMATICAL AND PHYSICAL SCIENCES

Año: 1998 vol. 98 p. 159 - 171

0035-8975

For a Banach space $E$ we define the class ${\cal P}_K (^NE)$of $K$-bounded $N$-homogeneous polynomials, where $K$ is a bounded subset of$E^{\prime}$. We investigate properties of $K$ which relate the space ${\cal %P}_K (^NE)$ with usual subspaces of ${\cal P} (^NE)$. We prove that $K$%-bounded polynomials are approximable when $K$ is a compact set where theidentity can be uniformly approximated by finite rank operators. The same istrue when $K$ is contained in the absolutely convex hull of a weakly nullbasic sequence of $E^{\prime}$. Moreover, in this case we prove that every $K$-bounded polynomial is extendible to any larger space.