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We investigate the block numerical range of bounded linear operators on arbitrary Banach spaces. We show that the spectrum of an operator is always contained in the closure of its block numerical range. The inclusion between block numerical ranges for refined block decompositions hold only in special cases which we characterize completely. Thereby we achieve a new characterization of Lp -spaces. Finally we obtain an estimate of the resolvent in terms of the block numerical range. All our results are new even for n×n -matrices. Mathematics subject classification (2010): 47A10, 47A12, 47A75, 47B65, 15A60, 46B08, 46B42.

On the block numerical range of operators on arbitrary Banach spaces