Strong operator topology

In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form ${\displaystyle T\mapsto \|Tx\|}$, as x varies in H.

Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map ${\displaystyle (T,x)\mapsto Tx}$ (taking values in H) is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets ${\displaystyle U(T_{0},x,\epsilon )=\{T:\|Tx-T_{0}x\|<\epsilon \}}$ (where T₀ is any bounded operator on H, x is any vector and ε is any positive real number).

In concrete terms, this means that ${\displaystyle T_{i}\to T}$ in the strong operator topology if and only if ${\displaystyle \|T_{i}x-Tx\|\to 0}$ for each x in H.

The SOT is stronger than the weak operator topology and weaker than the norm topology.

The SOT lacks some of the nicer properties that the weak operator topology has, but being stronger, things are sometimes easier to prove in this topology. It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.

The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus.

The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the WOT. Because of this, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.

This language translates into convergence properties of Hilbert space operators. For a complex Hilbert space, it is easy to verify by the polarization identity, that Strong Operator convergence implies Weak Operator convergence.

See also

• Strongly continuous semigroup
• Topologies on the set of operators on a Hilbert space

References

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Pedersen, Gert (1989). Analysis Now. Springer. ISBN 0-387-96788-5.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.