Best approximation and quasitriangular algebras

Abstract

If P \mathcal {P} is a linearly ordered set of projections on a Hilbert space and K \mathcal {K} is the ideal of compact operators, then Alg P + K \operatorname {Alg}\, \mathcal {P} + \mathcal {K} is the quasitriangular algebra associated with P \mathcal {P} . We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given T T and P \mathcal {P} , is there an A A in Alg P + K \operatorname {Alg}\, \mathcal {P} + \mathcal {K} such that ‖ T − A ‖ = inf { ‖ T − S ‖ : S ∈ Alg P + K } \left \| {T - A} \right \| = \inf \{ \left \| {T - S} \right \|:S \in \operatorname {Alg}\,\mathcal {P} + \mathcal {K}\} ? We prove that if A \mathcal {A} is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition Δ \Delta , then every operator T T has a best approximant in A + K \mathcal {A} + \mathcal {K} . We also show that if E \mathcal {E} is an increasing sequence of finite rank projections converging strongly to the identity then Alg E \operatorname {Alg}\,\mathcal {E} satisfies the condition Δ \Delta . Also, we show that if T T is not in Alg E + K \operatorname {Alg}\,\mathcal {E} + \mathcal {K} then the best approximants in Alg E + K \operatorname {Alg}\,\mathcal {E} + \mathcal {K} to T T are never unique.