Operator Topologies - Introduction

Introduction

Let {Tn} be a sequence of linear operators on the Hilbert space H. Consider the statement that Tn converges to some operator T in H. This could have several different meanings:

  • If, that is, the operator norm of Tn - T (the supremum of, where x ranges over the unit ball in H) converges to 0, we say that in the uniform operator topology.
  • If for all x in H, then we say in the strong operator topology.
  • Finally, suppose in the weak topology of H. This means that for all linear functionals F on H. In this case we say that in the weak operator topology.

All of these notions make sense and are useful for a Banach space in place of the Hilbert space H.

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