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A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded ) if every neighborhood of the origin absorbs it.
Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, [1] or, equivalently, a surjective isometry. [2] An equivalent definition is the following: Definition 2. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: U is surjective, and
Commutator subspaces, though, received sparse attention until the 1970s. American mathematician Paul Halmos in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators. [1]
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
For example, if X is a Banach space and T is a strictly singular operator in B(X) then its spectrum satisfies the following properties: (i) the cardinality of () is at most countable; (ii) () (except possibly in the trivial case where X is finite-dimensional); (iii) zero is the only possible limit point of (); and (iv) every nonzero () is an ...
An important class of examples is provided by Hilbert–Schmidt integral operators. Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator. The identity operator on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional.