The product of operators with closed range in Hilbert C*-modules

Abstract

Suppose $T$ and $S$ are bounded adjointable operators with close range between Hilbert C*-modules, then $TS$ has closed range if and only if $Ker(T)+Ran(S)$ is an orthogonal summand, if and only if $Ker(S^{\ast })+Ran(T^{\ast })$ is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between $Ran(S)$ and $Ker(T)\cap [Ker(T)\cap Ran(S){]}^{\perp }$ is positive and $\overline{Ker(S^{\ast })+Ran(T^{\ast })}$ is an orthogonal summand then $TS$ has closed range.

12 pages, abstract was changed, accepted

References in corpus (3)

• Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
• Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem
• Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^{\ast }$-modules

Cited by in corpus (7)

• Semi-Fredholm Theory on Hilbert C*-modules
• EP modular operators and their products
• On operators with closed range and semi-Fredholm operators over W*-algebras
• On compressions and generalized spectra of operators over C*-algebras
• The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules
• Some examples of EP operators
• The minimal and maximal symmetries for $J$-contractive projections