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The linear operators from to form a vector space for one writes over with respect to addition and multiplication by a scalar, given by the formulas. Multiplication composition of linear operators and is defined only for as the successive application of and. With respect to these three operations is an example of an associative algebra over with identity cf. Associative rings and algebras. Vector spaces over a fixed field objects and linear operators morphisms form, together with the composition law, the category.

The following concepts are special cases in connection with of general categorical concepts. For a linear operator its kernel or null-space is the subspace , its image is the subspace , and its cokernel is the quotient space. The rank of a linear operator is the dimension of its kernel and the nullity is the dimension of its kernel. A linear operator is called a monomorphism if and an epimorphism if. A linear operator is called a left respectively, right inverse of if is the identity in respectively, is the identity in.

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A linear operator that is simultaneously the left and right inverse of is called the inverse of. A linear operator respectively, endomorphism that has an inverse is called an isomorphism respectively, automorphism. The category is an Abelian category with respect to addition of linear operators; in particular, a linear operator that is a monomorphism and an epimorphism is an isomorphism.

Moreover, in every monomorphism has a left inverse and every epimorphism has a right inverse. By analogy with one introduces the categories and ; the objects of the first are Banach spaces, and the objects of the second are Hilbert spaces; the morphisms in both are continuous linear operators.

Hilbert space - Wikipedia

Both categories are additive cf. Additive category , but not Abelian. Isomorphisms in them are called topological isomorphisms; these are linear operators that have a continuous inverse. One of the most important typical problems of the "intrinsic" theory of linear operators is the problem of classifying endomorphisms or at least certain classes of them with respect to some equivalence or other.

For linear operators in pure algebra one considers, as a rule, the general categorical equivalence of endomorphisms in ; it is called similarity.

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In other words, linear operators and acting in and , respectively, are similar if for some isomorphism the diagram. Equivalence of continuous linear operators in general Banach spaces is called topological equivalence, and is understood in the general categorical sense, this time in ; this implies that the diagram 1 commutes for some topological isomorphism.

For linear operators in Hilbert spaces one chooses unitary equivalence of and as basis, corresponding to the requirement that the diagram 1 commutes for some unitary see below operator. Besides this, in the theory of linear operators between spaces with a topology there are important problems of approximating various classes of linear operators by operators of a comparatively simple structure. A significant role is played by problems about finding the general form of linear operators on concrete spaces, most frequently function spaces.

Among the invariants of similarity the most important are the spectrum and the number of invariant subspaces of given dimension. Let be a linear operator on. Its spectrum is the subset in consisting of those for which has no inverse. A subspace of is said to be invariant with respect to if implies that. Besides the kernel and the image of a linear operator, examples are the one-dimensional subspaces containing the so-called eigen vectors of the operator, that is, those , , for which ,.

An Introduction to Hilbert Spaces

The element , called an eigen value of , automatically belongs to 's spectrum. The concept of a linear operator is a special case of the concept of a morphism of modules, which is obtained by replacing the field by an arbitrary ring. In many respects morphisms of modules are not like linear operators in their properties, but results about the latter form one of the stimuli for studying them. The main analytic apparatus for such linear operators is the matrix notation. Let and be spaces with fixed bases , , and , , let be a linear operator, and let be the -th coefficients of the expansion of with respect to the second basis.

Then the -dimensional matrix is called the matrix of the linear operator in the bases and. If the matrix notation is usually taken in coincident bases that is, ,.

Subdiagonalization of operators in Hilbert space with compact imaginary part

By transfer to other bases the matrix of a linear operator is changed by simple formulas. To various characteristics of linear operators correspond, as a rule, effectively computable characteristics of their matrices. For example, and , where is the rank of the matrix; in particular, is an isomorphism if and only if ; this condition is equivalent to the determinant of being non-zero. To algebraic operations on linear operators correspond operations of the same name on their matrices, taken in fixed bases.

Linear operators are similar if and only if they can be described each in its "own" basis by the same matrix. The eigen values of a linear operator are the roots of the characteristic polynomial of its matrix. This implies that every linear operator on a finite-dimensional space over an algebraically closed field for example, has at least one eigen vector.

The spectrum of a linear operator on a finite-dimensional space over an arbitrary field is the set of its eigen values.

Derivation and Numerical analysis of an Attenuation Operator for non-relativistic waves

The problem of classifying endomorphisms of finite-dimensional spaces over an algebraically closed field has been completely solved, and the similarity classes have been described in terms of the invariant subspaces. The main step here is the following classification theorem, discovered by C. Jordan who considered the field. A matrix with a fixed on the main diagonal, ones immediately above it and zeros elsewhere, is called a Jordan block, and a block matrix with Jordan blocks of different dimensions and with different on the main diagonal and zeros elsewhere is called a Jordan matrix.

Then every endomorphism can be written in some basis as a Jordan matrix. The search for such a basis is called reduction of a linear operator to Jordan form. Since linear operators described by Jordan matrices are similar if and only if their matrices coincide up to a permutation of the Jordan blocks, the theorem means that Jordan matrices present, after suitable identification, a complete set of similarity invariants. A linear operator between Banach spaces is continuous if and only if it is bounded, that is, the image of every bounded set in is bounded in , or equivalently, if there is a finite number , called the operator norm a similar assertion is also true for arbitrary normed spaces.

The continuous linear operators from into form a subspace of which is a Banach space with respect to. The subspace , often denoted by , is even a Banach algebra with respect to operator multiplication. The class of these algebras is universal in the sense that every Banach algebra is topologically isomorphic to a subalgebra of for some. The subset of consisting of topological isomorphisms is open and for contains the open ball of unit radius with centre at.

A fundamental role in the "Banach" theory of linear operators is played by two theorems, which together with the Hahn—Banach theorem see also Linear functional are called the three fundamental principles of linear analysis. The simplest form of the "principle of uniform boundedness" is the Banach—Steinhaus theorem: If continuous linear operators , are such that for every one has , then.

The simplest form of the "open mapping principle" is Banach's theorem: If a continuous linear operator has an inverse, then this inverse operator is automatically continuous. The requirement that the spaces be "Banach" complete spaces is essential in both theorems. The spectrum of a continuous linear operator , usually considered for complex spaces, is a non-empty compact set in. In the case of an infinite-dimensional space the set of eigen values of a linear operator, called the point spectrum, generally forms only part of.

Read more. Theory of Linear Operators in Hilbert Space. Linear Systems and Operators in Hilbert Space. Spectral theory of operators in Hilbert space. Linear operators in Hilbert spaces.

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  • Structure of Hilbert Space Operators. Means of Hilbert Space Operators. Hilbert Space Operators. Structure of Hilbert space operators. Hilbert Space Operators in Quantum Physics. Commuting Nonselfadjoint Operators in Hilbert Space. Reine Angew. Kharkoff [Zapiski Inst. Paris, Ser. Control Optim. Roumaine Math. Pures Appl. Export Citation. User Account Log in Register Help. Search Close Advanced Search Help. My Content 1 Recently viewed 1 Isomorphic Schauder de Show Summary Details. More options …. Open Access.

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