On the topological boundary of the one-sided spectrum

It is well-known that the topological boundary of the spectrum of an operator is contained in the approximate point spectrum. We show that the one-sided version of this result is not true. This gives also a negative answer to a problem of Schmoeger. Denote by L(X) the algebra of all bounded linear operators acting in a Banach space X. For T ∈ L(X) denote by σ(T ), σl(T ) and σπ(T ) the spectrum, left spectrum and the approximate point spectrum of T , respectively: σ(T ) = {λ ∈ C : T − λ is not invertible}, σl(T ) = {λ ∈ C : T − λ is not left invertible}, σπ(T ) = {λ ∈ C : T − λ is not bounded below}. It is well-known that ∂σ(T ) ⊂ σπ(T ) ⊂ σl(T ) ⊂ σ(T ). This implies in particular that the outer topological boundaries (= the boundaries of the polynomially convex hull) of σ(T ), σl(T ) and σπ(T ) coincide. The aim of this paper is to show that the inner topological boundaries of σl and σπ can be different. The author wishes to express his thanks to G. Pisier for the proof of Proposition 3. We use the following notations. If X is a closed subspace of a Banach space Y then we denote c(X, Y ) = inf{‖P‖ : P ∈ L(Y ) is a projection with range X} (if X is not complemented in Y then we set c(X,Y ) = ∞). For Banach spaces X and Y denote by X⊗̂Y and X⊗̌Y the projective and injective tensor products (see [2]). Thus X⊗̂Y and X⊗̌Y are the completions of the algebraic tensor product X ⊗ Y endowed with the projective (injective) norms ‖u‖X⊗̂Y = inf {∑ i ‖xi‖ · ‖yi‖ : u = ∑