Linear operator

The term linear operator was introduced in functional analysis (a branch of mathematics) and is synonymous with the term linear mapping . A linear mapping is a structure-preserving mapping between vector spaces over a common body . If vector spaces are considered over the field of real or complex numbers and if these are provided with a topology ( locally convex spaces , normalized spaces , Banach spaces ), one preferably speaks of linear operators.

In contrast to finite-dimensional spaces, where linear operators are always restricted , unlimited linear operators also appear in infinite-dimensional spaces .

definition

Linear operator

Let and real or complex vector spaces. A mapping from to is called a linear operator if the following conditions hold for all and (or ): ${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle T}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle x, y \ in X}$${\ displaystyle \ lambda \ in \ mathbb {R}}$${\ displaystyle \ lambda \ in \ mathbb {C}}$

1. ${\ displaystyle T}$ is homogeneous: ${\ displaystyle T (\ lambda x) = \ lambda T (x)}$
2. ${\ displaystyle T}$ is additive: ${\ displaystyle T (x + y) = T (x) + T (y)}$

Antilinear operator

Be and complex vector spaces. An operator of in is called an anti-linear operator if all and the following conditions hold: ${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle T}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle x, y \ in X}$${\ displaystyle \ lambda \ in \ mathbb {C}}$

1. ${\ displaystyle T}$ is antihomogeneous: ${\ displaystyle T (\ lambda x) = {\ overline {\ lambda}} T (x)}$
2. ${\ displaystyle T}$ is additive: ${\ displaystyle T (x + y) = T (x) + T (y)}$

Examples

Linear operators

• Let it be a real matrix. Then the linear mapping is a linear operator of in .${\ displaystyle A}$${\ displaystyle n \ times m}$${\ displaystyle A \ colon x \ mapsto Ax}$${\ displaystyle \ mathbb {R} ^ {m}}$${\ displaystyle \ mathbb {R} ^ {n}}$
• The set of linear operators between two fixed vector spaces becomes a vector space itself through the definition of addition and scalar multiplication .${\ displaystyle (S + T) (x): = S (x) + T (x)}$ ${\ displaystyle (\ lambda S) (x): = \ lambda S (x)}$
• The derivative operator that assigns its derivative to a function is a linear operator.${\ displaystyle D \ colon C ^ {1} \ to C}$${\ displaystyle f \ mapsto Df = f '}$
• Let be two real numbers. The operator that assigns a real number to an integrable function is linear.${\ displaystyle a ${\ displaystyle \ textstyle f \ mapsto \ int _ {a} ^ {b} f (x) \ mathrm {d} x}$
• Every linear functional on a vector space is a linear operator.

Antilinear operator

• If there is a complex Hilbert space and its dual space , then according to the representation theorem of Fréchet-Riesz there is exactly one for each , so that applies to all . The figure is anti-linear. This is due to the fact that a complex scalar product is anti-linear in the second variable.${\ displaystyle (H, \ langle \ cdot, \ cdot \ rangle _ {H})}$${\ displaystyle H \, '}$${\ displaystyle f \ in H \, '}$${\ displaystyle y_ {f} \ in H}$${\ displaystyle f (x) = \ langle x, y_ {f} \ rangle _ {H}}$${\ displaystyle x \ in H}$${\ displaystyle H \, '\ rightarrow H, f \ mapsto y_ {f}}$${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$

Importance and uses

The importance of linear operators is that they respect the linear structure of the underlying space, i.e. that is, they are homomorphisms between vector spaces.

Applications of linear operators are:

• In four-pole theory ( electrical engineering ), the relationships between the input variables (current strength and voltage) and the output variables (current strength and voltage) are viewed as mutually linearly dependent on one another. The dependencies can be described by 2 × 2 matrices.

Constrained linear operators

Definitions

Let and be two normalized vector spaces and a linear operator. The operator norm of is defined by ${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle A \ colon V \ to W}$${\ displaystyle A}$

${\ displaystyle \ | A \ |: = \ inf \ {M \ geq 0, \; \ | Ax \ | _ {W} \ leq M \ | x \ | _ {V} {\ text {for all}} x \ in V \}}$,

where for this constant

${\ displaystyle \ | A \ | = \ sup _ {x \ in V, \; x \ neq 0} {\ frac {\ | Ax \ | _ {W}} {\ | x \ | _ {V}} } = \ sup _ {\ | x \ | _ {V} \ leq 1} \ | Ax \ | _ {W} = \ sup _ {\ | x \ | _ {V} = 1} \ | Ax \ | _ {W}}$

applies. If the operator norm is finite, the operator is called bounded, otherwise unbounded.

The set of all bounded linear operators from normalized space into normalized space is called . With the operator norm, this is itself a normalized vector space. If complete , it is even a Banach space . If with is identical, it is also written in abbreviated form. The bounded linear operators can be characterized as follows: ${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle {\ mathfrak {L}} (V, W)}$${\ displaystyle W}$ ${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle {\ mathfrak {L}} (V)}$

If a linear operator is from to , then the following statements are equivalent: ${\ displaystyle T}$${\ displaystyle V}$${\ displaystyle W}$

1. ${\ displaystyle T}$is restricted, d. H. in included.${\ displaystyle {\ mathfrak {L}} (V, W)}$
2. ${\ displaystyle T}$is evenly steady on .${\ displaystyle V}$
3. ${\ displaystyle T}$is continuous at every point of .${\ displaystyle V}$
4. ${\ displaystyle T}$is continuous at a point of .${\ displaystyle V}$
5. ${\ displaystyle T}$is steadily in .${\ displaystyle 0 \ in V}$

Examples of bounded linear operators

• ${\ displaystyle I_ {V} \ in {\ mathfrak {L}} (V)}$with , where the identical operator is on .${\ displaystyle \ | I_ {V} \ | = 1}$${\ displaystyle I_ {V}}$${\ displaystyle V}$
• ${\ displaystyle P \ in {\ mathfrak {L}} (H)}$with , where is an orthogonal projection on the Hilbert space .${\ displaystyle \ | P \ | = 1}$${\ displaystyle P \ neq 0}$ ${\ displaystyle H}$
• ${\ displaystyle (n_ {k}) \ in {\ mathfrak {L}} (l_ {p})}$with , wherein the sequence is limited, and as a diagonal operator on the sequence space with is interpreted.${\ displaystyle \ textstyle \ | (n_ {k}) \ | = \ max _ {k} | n_ {k} |}$${\ displaystyle (n_ {k})}$ ${\ displaystyle l_ {p}}$${\ displaystyle 1 \ leq p \ leq \ infty}$
• The shift operator is constrained with , where is defined on the sequence space with .${\ displaystyle S \ in {\ mathfrak {L}} (l_ {p})}$${\ displaystyle \ | S \ | = 1}$${\ displaystyle S ((x_ {1}, x_ {2}, x_ {3}, \ dotsc)): = (0, x_ {1}, x_ {2}, x_ {3}, \ dotsc)}$${\ displaystyle l_ {p}}$${\ displaystyle 1 \ leq p \ leq \ infty}$
• Let it be a compact set and the Banach space of continuous functions with the supremum norm . Further suppose that the linear operator is defined by for . Then is and .${\ displaystyle K}$${\ displaystyle {\ mathfrak {C}} (K)}$${\ displaystyle K}$${\ displaystyle f \ in {\ mathfrak {C}} (K)}$${\ displaystyle T_ {f} \ colon {\ mathfrak {C}} (K) \ rightarrow {\ mathfrak {C}} (K)}$${\ displaystyle T_ {f} (g) (k): = (fg) (k)}$${\ displaystyle k \ in K}$${\ displaystyle T_ {f} \ in {\ mathfrak {L}} ({\ mathfrak {C}} (K))}$${\ displaystyle \ | T_ {f} \ | = \ | f \ | _ {\ infty}}$
• Let be a measure space and the L p -space of the equivalence classes of the measurable functions integrable in -th power on with the L p -norm for . Further let and the linear operator be defined by for . Then is and .${\ displaystyle \ lbrack X, {\ mathfrak {B}}, \ mu \ rbrack}$${\ displaystyle L_ {p} = L_ {p} (X, {\ mathfrak {B}}, \ mu)}$${\ displaystyle p}$${\ displaystyle X}$${\ displaystyle 1 \ leq p \ leq \ infty}$${\ displaystyle f \ in L _ {\ infty}}$${\ displaystyle T_ {f} \ colon L_ {p} \ to L_ {p}}$${\ displaystyle T_ {f} (g) (x): = (fg) (x)}$${\ displaystyle x \ in X}$${\ displaystyle T_ {f} \ in {\ mathfrak {L}} (L_ {p})}$${\ displaystyle \ | T_ {f} \ | = \ | f \ | _ {\ infty}}$

Applications

• Spectral theory
• Functional calculus , d. H. for a bounded, real or complex-valued measurable function and a bounded linear operator can be defined.${\ displaystyle f}$${\ displaystyle T}$${\ displaystyle f (T)}$

Unconstrained linear operators

When considering unrestricted linear operators, operators are often also allowed whose domain is only a subspace of the space under consideration; for example, one speaks of unbounded linear operators on Hilbert spaces, so a prehilbert space is also allowed as a domain as a subspace of a Hilbert space, more precisely one then speaks of densely defined unbounded linear operators (see below). The operator is understood as a partial mapping .

An operator is said to be densely defined if its domain is a dense subset of the initial space . The interest in unbounded operators is based on the investigation of differential operators and their spectrum of eigenvalues and observable algebras .

A large class of unbounded linear operators are the closed operators . These are operators whose graph is closed in the product topology of . For closed operators, e.g. B. the spectrum can be defined. ${\ displaystyle A \ colon V \ rightarrow W}$${\ displaystyle \ Gamma (A): = \ {(\ phi, A \ phi): \ phi \ in D \}}$${\ displaystyle V \ times W}$

The theory of the unbounded operators was established by John von Neumann in 1929. In 1932, independently of von Neumann, Marshall Harvey Stone developed the theory of unbounded operators.

example

Consider the differential operator on the Banach space of continuous functions on the interval . If one chooses the once continuously differentiable functions as the domain , then it is a closed operator that is not restricted. ${\ displaystyle Af: = f '\,}$${\ displaystyle C [a, b]}$${\ displaystyle [a, b]}$${\ displaystyle {\ mathcal {D}} (A)}$${\ displaystyle {\ mathcal {D}} (A): = C ^ {1} [a, b]}$${\ displaystyle A}$

Applications

• Differential and multiplication operators are i. A. unlimited.
• The representation of observables in quantum mechanics requires unlimited linear operators, since the operators assigned to the observables i. A. are unlimited.

Convergence terms / topologies on operator spaces

If the underlying vector space is finite-dimensional with dimension , then a vector space is the dimension . In this case, all norms are equivalent , that is, they provide the same concept of convergence and the same topology . ${\ displaystyle n}$${\ displaystyle L (V)}$${\ displaystyle n ^ {2}}$

In the infinite-dimensional, on the other hand, there are various non-equivalent topologies. Now be and Banach spaces and a sequence (or a network ) in . ${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle (T_ {i}) _ {i \ in I}}$${\ displaystyle L (E, F)}$

Standard topology

${\ displaystyle T_ {i}}$converges in the norm topology to if and only if: ${\ displaystyle T}$

${\ displaystyle \ lim _ {i} \ | T-T_ {i} \ | = 0}$

The standard topology is the topology represented by the open spheres produced is.

Strong operator topology

${\ displaystyle T_ {i}}$converges in the strong operator topology ( stop for short ) to if and only if it converges point by point: ${\ displaystyle T}$

${\ displaystyle \ lim _ {i} T_ {i} x = Tx \ quad \ forall x \ in E}$

or in other words:

${\ displaystyle 0 = \ lim _ {i} \ | T_ {i} x-Tx \ | = \ lim _ {i} \ | (T_ {i} -T) x \ | \ quad \ forall x \ in E }$

The associated topology is the initial topology , which is defined by the set of linear mappings

${\ displaystyle \ left \ lbrace \ left. {\ begin {matrix} L (E, F) & \ to & F \\ T & \ mapsto & Tx \ end {matrix}} \, \ right | \, x \ in E \ right \ rbrace}$

is produced. This is the smallest topology in which all of these maps are continuous. with the strong operator topology there is therefore a locally convex space . ${\ displaystyle L (E, F)}$

Alternatively expressed: The strong operator topology is the product topology of all functions from to , restricted to the (possibly restricted) linear operators. ${\ displaystyle E}$${\ displaystyle F}$

Weak operator topology

${\ displaystyle T_ {i}}$converges in the weak operator topology to if and only if ${\ displaystyle T}$

${\ displaystyle \ lim _ {i} \ varphi (T_ {i} x) = \ varphi (Tx) \ quad \ forall x \ in E, \ varphi \ in F ^ {*}}$

or in other words:

${\ displaystyle \ lim _ {i} | \ varphi (T_ {i} x-Tx) | = 0 \ quad \ forall x \ in E, \ varphi \ in F ^ {*}}$

(Here denotes the continuous dual space of F) ${\ displaystyle F ^ {*}}$

The associated topology is the initial topology , which is defined by the set of linear functionals

${\ displaystyle \ left \ lbrace \ left. {\ begin {matrix} L (E, F) & \ to & \ mathbb {C} \\ T & \ mapsto & \ varphi (Tx) \ end {matrix}} \, \ right | x \ in E, \ varphi \ in F ^ {*} \ right \ rbrace}$

is produced. This is the smallest topology in which all of these functionals are continuous. with the weak operator topology, there is also a locally convex space . ${\ displaystyle L (E, F)}$

literature

• Hans Wilhelm Alt: Linear Functional Analysis. An application-oriented introduction. 5th edition. Springer-Verlag, 2006, ISBN 3-540-34186-2 .

Individual evidence

1. Dirk Werner : Functional Analysis. 7th, corrected and enlarged edition. Springer, 2011. ISBN 978-3-642-21016-7 . Sentence II.1.4.
2. Dirk Werner: Functional Analysis. 7th, corrected and enlarged edition. Springer, 2011. ISBN 978-3-642-21016-7 . Chapter VII.6.