Search results
Results from the WOW.Com Content Network
Let and be Hilbert spaces, and let : be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, is dense in . Let : denote the adjoint of .
Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.
If the C*-algebra is the algebra of all bounded operators on a Hilbert space , then the bounded observables are just the bounded self-adjoint operators on . If v {\displaystyle v} is a unit vector of H {\displaystyle \mathbb {H} } then ω ( A ) = v , A v {\displaystyle \omega (A)=\langle v,Av\rangle } is a state on the C*-algebra, meaning the ...
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
The conjecture is true if the Hilbert space is not separable (i.e. if it has an uncountable orthonormal basis). In fact, if x {\displaystyle x} is a non-zero vector in H {\displaystyle H} , the norm closure of the linear orbit [ x ] {\displaystyle [x]} is separable (by construction) and hence a proper subspace and also invariant.
Learn how to download and install or uninstall the Desktop Gold software and if your computer meets the system requirements.
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.