Operators with the Kato property on Banach spaces
This work studies a class of bounded linear operators between Banach spaces, called operators with the Kato property, which includes strictly singular operators. The main result shows that if T is a dense-range operator with the Kato property and E has a separable quotient, then for each proper dens...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | RUIdeRA. Repositorio Institucional de la UCLM |
| OAI Identifier: | oai:ruidera.uclm.es:10578/47708 |
| Acceso en línea: | https://hdl.handle.net/10578/47708 |
| Access Level: | acceso abierto |
| Palabra clave: | Banach space Operator with Kato property Proper dense operator range Quasicomplemented subspace Separable quotient |
| Sumario: | This work studies a class of bounded linear operators between Banach spaces, called operators with the Kato property, which includes strictly singular operators. The main result shows that if T is a dense-range operator with the Kato property and E has a separable quotient, then for each proper dense operator range R(T) there exists a closed subspace X such that E/X is separable, T(X) is dense in F, and X has infinite codimension. If F is weak*-separable, X can be chosen to satisfy an additional structural condition.The paper applies these findings to the geometry of Banach spaces, extending results by Johnson and Plichko. In particular, if X and Y are quasicomplemented but not complemented subspaces of a Banach space E, and X has a separable quotient, then X contains a closed subspace X0 such that E/X0 is separable and X0 is a quasicomplement of Y. Similarly, if T is an operator with non-closed range and E has a separable quotient, there exists a weak*-closed subspace X0 such that T(X0) is dense. Refinements are obtained when the range is weak*-separable.Finally, the authors show that if E has a separable quotient, then the dual space E* is weak*-separable if and only if, for every closed subspace X and every proper dense operator range containing X, one can find a quasicomplement Y of X such that the corresponding quotient is separable. |
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