EQUIVALENT NORMS ON FOCK SPACES WITH SOME APPLICATION TO EXTENDED CESARO OPERATORS

Let F p. be the Fock space of all holomorphic functions f in C-n with the Fock norm parallel to f parallel to(p,gamma) = {integral(Cn)vertical bar f(z)e(-gamma vertical bar z vertical bar 2/2)vertical bar(p) dA(z)}(1/p) < infinity, where p,gamma are positive numbers. We prove that, given any positive integer m, the Fock norm parallel to f parallel to(p,gamma) is equivalent to Sigma(vertical bar alpha vertical bar <= m-1) vertical bar partial derivative(alpha)f(0)vertical bar + {Sigma(vertical bar alpha vertical bar=m)integral(Cn) vertical bar partial derivative(alpha)f(z)(1 + vertical bar z vertical bar)(-m)e (gamma vertical bar z vertical bar 2/2)vertical bar(p) dA(z)}(1/p). As some application we characterize these holomorphic functions g in C-n for which the induced extended Cesaro operator T-g is bounded (or compact) from one Fock space F-gamma(p) to another F-gamma(q).