Abstract
We show that the distance from an operator T to the normal operators changes drastically when we increase the multiplicity to ∞. If T(∞) denotes a countably in nite direct sum of copies of T, and if C is the constant in the remarkable distance estimate of Kachkovskiy and Safarov (2016), then the distance from T(∞) to the set of normal operators is at most 2C||T*T-TT*||1/2. There is no such relation for the in mum of the distances from T(n) (1≤n<∞) to the normals. We use this to show that if H is a nonseparable Hilbert space and K is any closed ideal in B(H) that is not the ideal of compact operators, then any normal element of B(H)=K can be lifted to a normal element of B(H).
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