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Abstract. We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let T, A ∈ B(X, Y), where X is a real Banach space and Y is a real normed linear space. We find sufficient condition for T ⊥ B A ⇔ T x⊥ B Ax for some x ∈ S X with T x = T , and use it to show that T is a smooth point in B(X, Y) if T attains its norm at unique (upto muliplication by scalar) vector x ∈ S X , T x is a smooth point of Y and sup y∈C T y < T for all closed subsets C of S X with d(±x, C) > 0. For operators on a Hilbert space H we show that T ⊥ B A ⇔ T x⊥ B Ax for some x ∈ S H with T x = T if and only if the norm attaining set M T = {x ∈ S H : T x = T } = S H 0 for some finite dimensional subspace H 0 and T Ho ⊥ < T . We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.

We present some upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improves on the existing upper and lower bounds. We also present an upper bound for the spectral radius of sum of product of n pairs of operators. As an application of the results obtained, we provide a better estimation for the zeros of a given polynomial.2010 Mathematics Subject Classification. Primary 47A12, 15A60, 26C10.