TY - JOUR

T1 - Equivalence after extension for compact operators on Banach spaces

AU - van der Horst, S.

AU - Messerschmidt, M.

AU - Ran, A.C.M.

N1 - PT: J; NR: 18; TC: 0; J9: J MATH ANAL APPL; PG: 14; GA: CM1KZ; UT: WOS:000357441100010

PY - 2015

Y1 - 2015

N2 - In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In the present paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions for compact Hilbert space operators to be equivalent after extension, in terms of their singular values, we prove, under certain additional conditions, the necessity of a similar relationship between the s-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary s-functions. We investigate equivalence after extension for operators on ℓp-spaces. We show that two operators that act on different ℓp-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension; however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces X and Y, we investigate consequences of an operator on X being equivalent after extension to a compact operator on Y. We show that, in this case, a closed finite codimensional subspace of Y must embed into X, and that certain general Banach space properties must transfer from X to Y. We also show that no operator on X can be equivalent after extension to an operator on Y, if X and Y are essentially incomparable Banach spaces.

AB - In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In the present paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions for compact Hilbert space operators to be equivalent after extension, in terms of their singular values, we prove, under certain additional conditions, the necessity of a similar relationship between the s-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary s-functions. We investigate equivalence after extension for operators on ℓp-spaces. We show that two operators that act on different ℓp-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension; however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces X and Y, we investigate consequences of an operator on X being equivalent after extension to a compact operator on Y. We show that, in this case, a closed finite codimensional subspace of Y must embed into X, and that certain general Banach space properties must transfer from X to Y. We also show that no operator on X can be equivalent after extension to an operator on Y, if X and Y are essentially incomparable Banach spaces.

U2 - 10.1016/j.jmaa.2015.05.059

DO - 10.1016/j.jmaa.2015.05.059

M3 - Article

SN - 0022-247X

VL - 431

SP - 136

EP - 149

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -