Operator theory Operator theory In mathematics, a Banach space is said to have the approximation property (AP in short), if every compact operator is a limit of finite rank operators. The converse is always true. ...more on Wikipedia about "Approximation property"
In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X, ...more on Wikipedia about "Bounded operator"
In functional analysis, the Calkin algebra is the quotient of B(H), the set of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. ...more on Wikipedia about "Calkin algebra"
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators. ...more on Wikipedia about "Closed operator"
In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous. Any L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite rank operators in an infinite-dimensional setting. When X = Y and is a Hilbert space, it is true that any compact operator is a limit of finite rank operators, so that the class of compact operators can be defined alternatively as the closure in the operator norm of the finite rank operators. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in the end Enflo gave a counter-example. ...more on Wikipedia about "Compact operator"
In mathematics, the composition operator of a given function g is defined as that operator which maps functions to functions as ...more on Wikipedia about "Composition operator"
In the theory of von Neumann algebras, a crossed product ...more on Wikipedia about "Crossed product"
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. ...more on Wikipedia about "Discrete Laplace operator"
The Fredholm alternative of functional analysis, initially found in Ivar Fredholm's work on integral equations, is a generalisation to certain contexts, in V has infinite dimension. ...more on Wikipedia about "Fredholm alternative"
In mathematics, a Fredholm determinant is a complex analytic function which generalizes the characteristic polynomial of a matrix. It is defined for those operators which have continuous kernels, i.e., kernels in the sense of mathematical analysis. The Front for the Math arXiv has several papers utilizing Fredholm determinants. ...more on Wikipedia about "Fredholm determinant"
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. Fredholm kernels are named for Ivar Fredholm. Much of the theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955. ...more on Wikipedia about "Fredholm kernel"
In mathematics, a Fredholm operator is a bounded linear operator between two Banach spaces whose range is closed and whose kernel and cokernel are finite-dimensional. Equivalently, an operator T : X → Y is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator ...more on Wikipedia about "Fredholm operator"
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. This result was a significant point in the development of the theory of C*-algebras in the early 1940s since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an algebra of operators. ...more on Wikipedia about "Gelfand–Naimark theorem"
The Hamiltonian, denoted H, has two distinct but closely related meanings. In classical mechanics, it is a function that describes the state of a mechanical system in terms of position and momentum variables (i.e. symplectic variables), which is the basis for a re-formulation of classical mechanics known as Hamiltonian mechanics. In quantum mechanics, the Hamiltonian is the observable corresponding to the total energy of a system. This article discusses the Hamiltonian operator in quantum mechanics. For other uses, see Hamiltonian (disambiguation). ...more on Wikipedia about "Hamiltonian (quantum mechanics)"
In mathematics, specifically in functional analysis, one associates to every linear operator on a Hilbert space its adjoint operator. ...more on Wikipedia about "Hermitian adjoint"
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces. Hilbert spaces are studied in functional analysis. ...more on Wikipedia about "Hilbert space"
In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H such that there exists an orthonormal basis of ...more on Wikipedia about "Hilbert-Schmidt operator"
In mathematics, an invariant subspace of a linear mapping over some vector space V is a subspace W of V such that T(W) is contained in W. ...more on Wikipedia about "Invariant subspace"
In mathematics, Kuiper's theorem is a result on the topology of the operators on an infinite-dimensional complex Hilbert space H. It states that the topological space X of all linear operators L from H to itself, which are bounded operators and invertible, is such that for any finite complex Y, there is just one homotopy class of mappings from Y to X. Here the topology on X is the norm topology of operators, and the single class must be that of constant mappings. It is a corollary, itself often called Kuiper's theorem, that X is contractible. ...more on Wikipedia about "Kuiper's theorem"
In mathematics, the min-max theorem or variational theorem is an important result in the theory of Hilbert spaces. ...more on Wikipedia about "Min-max theorem"
In functional analysis, nest algebras are a class of operator algebras which generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by John Ringrose in the mid- 1960s and have many interesting properties. They are non- selfadjoint algebras, are closed in the weak operator topology and are reflexive. ...more on Wikipedia about "Nest algebra"
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In mathematics, a nuclear operator is roughly a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace). ...more on Wikipedia about "Nuclear operator"
In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. Although important, nuclear spaces are not widely used, possibly because the definition is notoriously difficult to understand. ...more on Wikipedia about "Nuclear space"
In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. An observable is a self-adjoint operator. By extension, operator is also used to refer to an element of a C*-algebra. For the justification of this usage, see Gelfand–Naimark theorem ...more on Wikipedia about "Operator (physics)"
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. In particular, it is a set of operators with both algebraic and topological closure properties. Though operator algebras are studied in this generality in the research literature (for example, algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a Hilbert space. ...more on Wikipedia about "Operator algebra" Good to know shortopedia. shortopedia
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