Let H be an infinite dimensional complex Hilbert space and let f be a surjective linear map on B(H) with f(I)-IÎk(H), where k(H) denotes the closed ideal of all compact operators on H. If f preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then f is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

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The main purpose of this paper is to study a general norm on extension of a Hilbert’s type linear operator in the continuous and discrete form. In addition to expressing the norm of a Hilbert’s type linear operator T: L2 (0, ¥) ® L2 (0, ¥), a more general case with l > 0, for the continuous form has been studied. By putting l = 1 a norm of extension of Hilbert’s integral linear operator is obtained. Similar results have been expressed for series when 0 < l £ 2.

In this paper we obtain coefficient characterization, extreme points and distortion bounds for the classes of harmonic p-valent functions defined by certain modified operator. Some of our results improve and generalize previously known results.