### Approximation numbers for relatively bounded operators

#### Abstract

Let T be a densely defined closed operator between Banach spaces X and Y. Aconcept of approximation

numbers, called T–approximation numbers, is considered for T–bounded operators between Banach

spaces X and Z with their domains contained in X, and some properties of such T–approximation numbers

are studied. The theorems proved in the paper include a result on approximation of T–approximation numbers

of a T–bounded operator A using T–approximation numbers of An, where fAng is a certain sequence

of operators which converges to A in some sense. This result is analogous to a theorem proved recently by

the authors in [3] for bounded linear operators.

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