Best approximation and quasitriangular algebras

Abstract:

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