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.

Read more about this topic:  Operator Topologies

Famous quotes containing the word introduction:

    “Such is oftenest the young man’s introduction to the forest, and the most original part of himself. He goes thither at first as a hunter and fisher, until at last, if he has the seeds of a better life in him, he distinguishes his proper objects, as a poet or naturalist it may be, and leaves the gun and fish-pole behind. The mass of men are still and always young in this respect.”
    —Henry David Thoreau (1817–1862)

    “Do you suppose I could buy back my introduction to you?”
    —S.J. Perelman, U.S. screenwriter, Arthur Sheekman, Will Johnstone, and Norman Z. McLeod. Groucho Marx, Monkey Business, a wisecrack made to his fellow stowaway Chico Marx (1931)

    “We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.”
    —Angela Carter (1940–1992)