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A Banach space operator T ∈ B(X ) has a left m-inverse (resp., an essential left m-inverse) for some integer m ≥ 1 if there exists an operator S ∈ B(X ) (resp., an operator S ∈ B(X ) and a compact operator K ∈ B(X )) such thatis strictly left m-invertible (resp., strictly essentially left m-invertible), then T1 ⊗ T2 is: (i) left (m + n − 1)-invertible (resp., essentially left (m + n − 1)-invertible) if and only if T2 is left n-invertible (resp., essentially left n-invertible); (ii) strictly left (m + n − 1)invertible (resp., strictly essentially left (m + n − 1)-invertible) if and only if T2 is strictly left n-invertible (resp., strictly essentially left n-invertible).2010 Mathematics Subject Classification: Primary 47A80, 47A10; Secondary 47B47. Key words and phrases: Banach space, left n-invertible operator, essentially left ninvertible operator, tensor product, left-right multiplication operator.

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Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A • ∈ B(K) denote the Berberian extension of an operator A ∈ B(H). It is proved that the set theoretic function σ , the spectrum, is continuous on the setand A ii has SVEP for all 1 i m, then σ is continuous on C S (m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP.