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∗-Algebras in Several Complex Variables
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Recall that an associative complex Banach *-algebra A is called a C*-algebra if ||a*a|| = ||a||2 for all a ∈ A. The Banach *-algebra ℬ(H) of all bounded operators on a complex Hilbert space H is a C* -algebra, and so is every operator-norm closed *-subalgebra. Conversely, by the Gelfand-Naimark theorem, every C*-algebra can be realized in this way. The compact operatorsK(H) form a C* -subalgebra of ℬ(H).
Banach Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
If, in addition, T is continuous, then T is said to be completely continuous. If V is a Banach space (complete), then, according to Theorem 4.9.2, T is compact if and only if it maps bounded sets in U into totally bounded sets in V. This implies that every compact operator is bounded and, therefore, in particular, every compact linear operator is automatically completely continuous. Note also that, since in a finite-dimensional space boundedness is equivalent to the total boundedness, every bounded operator with a finite-dimensional range is automatically compact. In particular, every continuous linear operator with a finite-dimensional range is compact. This also implies that every linearT operator defined on a finite-dimensional space U is compact. Indeed, T is automatically continuous and the range of T is of finite dimension.
Bounded linear operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
A bounded linear transformation is usually referred to as a bounded operator, or simply as an operator. The set of bounded operators between two inner product spaces H and K is denoted by B(H, K), and the notation for the space B(H, H) is usually abbreviated to B(H).
The negative exponential transformation: a linear algebraic approach to the Laplace transform
Published in International Journal of Mathematical Education in Science and Technology, 2023
Laplace transforms appear in a vectorised form as the resolvent of operators in the theory of semigroups of operators. In this context, the resolvent set is an unbounded infinite half ray bounded from below by a suitable constant. In the theory of bounded operators, the spectrum is always bounded and the resolvent set is the unbounded complement.