On the exponential and approximate point spectrum of a bounded linear operator in Banach spaces

This paper investigates the exponential and approximate point spectrum of a bounded linear operator in Banach spaces. We define the operator exp(T) for a bounded linear operator T in a Banach space X and determine if exp(T) is invertible. We show that if S is a bounded linear operator in X and S commutes with T then exp(S + T) equals exp(T).exp(S). If H is a Hilbert space and T is a bounded linear operator in H which is normal such that T commutes with a bounded linear operator S in H, then S commutes with the adjoint of T. The converse of this statement holds. For a bounded linear operator A in H, the boundary of the spectrum of A is contained in subset of the approximate point spectrum of A. Also, if A and B are bounded linear operators in H which are similar, then A and B have the same spectrum, point spectrum, approximate point spectrum and compression spectrum. Using the Spectral Mapping Theorem, we have shown that the exponential and approximate point spectrum are independent of the Banach space.