With stronger assumptions, when " T " is a normal operator acting on a Hilbert space, the domain of the functional calculus can be broadened.
42.
Similarly, the continuous functional calculus holds for normal elements in any C *-algebra and the measurable functional calculus for normal elements in any von Neumann algebra.
43.
Similarly, the continuous functional calculus holds for normal elements in any C *-algebra and the measurable functional calculus for normal elements in any von Neumann algebra.
44.
Using the holomorphic functional calculus it can be checked that " D " 1 / 2 " B " 2 is a smoothing operator.
45.
This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via functional calculus ( for example, via the spectral theorem ).
46.
This explicit identification of the operators " e i " ( " T " ) in turn gives an explicit form of holomorphic functional calculus for matrices:
47.
More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function.
48.
In the wider context, multiplier operators are special cases of spectral multiplier operators, which arise from the functional calculus of an operator ( or family of commuting operators ).
49.
The more general continuous functional calculus can be defined for any self-adjoint ( or even normal, in the complex case ) bounded linear operator on a Hilbert space.
50.
However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of " T " as a multiplication operator.
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