Linear Operators in a Hilbert Space
The idea of a linear operator or transformation in a Hilbert space ℌ (or a Banach space) is a direct generalization of the idea of a linear transformation in a finite-dimensional space. One point, however, needs emphasis (mainly because it is sometimes ignored, especially in books on quantum mechanics), namely, an operator A cannot be regarded as fully specified until its domain of definition (i.e., the set of those x in ℌ for which Ax is meaningful) has been specified; operators with different domains of definition have to be regarded as different operators. It is customary to require the domain of definition to be a linear set (manifold) in ℌ, for the obvious reason that if A is linear and Ax is defined in a set S, then Ay can be uniquely defined, by linearity, when y is any finite linear combination of elements of S. However, further extensions are not generally unique, except in special circumstances.
KeywordsLinear operators or transformations in a Hilbert space domain range bound Extension Theorem Banach algebras adjoints symmetric, self-adjoint, and unitary operators integral and differential operators symmetric operators with no self-adjoint extension and ones with many simple Sturm-Liouville operators closed and closable operators the graph of an operator radial momentum operators
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