On a Topological Property of certain Calkin Algebras | Zendy

Meyer Michael J. | Zendy

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On a Topological Property of certain Calkin Algebras

Author(s) -

Meyer Michael J.

Publication year - 1992

Publication title -

bulletin of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.396

H-Index - 48

eISSN - 1469-2120

pISSN - 0024-6093

DOI - 10.1112/blms/24.6.591

Subject(s) - mathematics , bounded function , norm (philosophy) , quotient , bounded operator , linear operators , operator algebra , compact operator , ideal (ethics) , combinatorics , subspace topology , algebra over a field , discrete mathematics , pure mathematics , mathematical analysis , extension (predicate logic) , philosophy , epistemology , political science , computer science , law , programming language

Let X = 1 p , 1 ⩽ p < ∞, or X = c 0 , B ( X ) be the algebra of all bounded linear operators on X , H ( X ) be the ideal of compact operators in B ( X ), and C ( X ) = B ( X )/ H ( X ) be the Calkin algebra on X . For T ε B ( X ), let ∥ T ∥ c = dist( T , H ( X )) be the essential norm of T that is the norm of T + H ( X ) in C ( X ). It is shown that for any operator T ε B ( X ) and any number 0 < t < 1, there exists a closed infinite dimensional subspace Z Z ⊆ X such that ∥ Tx∥ ⩾ t ∥ T ∥ c , for all x ε Z . As a consequence, it is shown that every (not necessarily complete) submultiplicative norm on the Calkin algebra C ( X ) is equivalent to the quotient norm ∥ ∥ c on C ( X ).

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