Antilinear map explained

f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors x, y \in V and every complex number s, where \overline{s} denotes the complex conjugate of s.

Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity.

Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces.

Definitions and characterizations

A function is called or if it is additive and conjugate homogeneous. An on a vector space

V is a scalar-valued antilinear map.

A function

f is called if$$f(x\; +\; y)\; =\; f(x)\; +\; f(y)\; \backslash quad\; \backslash text\; x,\; y$$while it is called if $$f(ax)\; =\; \backslash overline\; f(x)\; \backslash quad\; \backslash text\; x\; \backslash text\; a.$$ In contrast, a linear map is a function that is additive and homogeneous, where f is called if$$f(ax)\; =\; a\; f(x)\; \backslash quad\; \backslash text\; x\; \backslash text\; a.$$

An antilinear map

f : V \to W may be equivalently described in terms of the linear map \overline{f} : V \to \overline{W} from V to the complex conjugate vector space \overline{W}.

Examples

Anti-linear dual map

Given a complex vector space

V of rank 1, we can construct an anti-linear dual map which is an anti-linear map $$l:V\; \backslash to\; \backslash Complex$$ sending an element x _{1} + iy _{1} for x _{1,y} _{1} \in \R to $$x\_1\; +\; iy\_1\; \backslash mapsto\; a\_1\; x\_1\; -\; i\; b\_1\; y\_1$$ for some fixed real numbers a _{1,b} _{1.} We can extend this to any finite dimensional complex vector space, where if we write out the standard basis e _{1,} \ldots, e _{n} and each standard basis element as $$e\_k\; =\; x\_k\; +\; iy\_k$$ then an anti-linear complex map to \Complex will be of the form $$\backslash sum\_k\; x\_k\; +\; iy\_k\; \backslash mapsto\; \backslash sum\_k\; a\_k\; x\_k\; -\; i\; b\_k\; y\_k$$ for a _{k,b} _{k} \in \R.

Isomorphism of anti-linear dual with real dual

The anti-linear dual^{[1] pg 36} of a complex vector space

V $$\backslash operatorname\_(V,\backslash Complex)$$ is a special example because it is isomorphic to the real dual of the underlying real vector space of V, Hom _{\R(V,\R).} This is given by the map sending an anti-linear map $$\backslash ell:\; V\; \backslash to\; \backslash Complex$$to $$\backslash operatorname(\backslash ell)\; :\; V\; \backslash to\; \backslash R$$ In the other direction, there is the inverse map sending a real dual vector $$\backslash lambda\; :\; V\; \backslash to\; \backslash R$$ to $$\backslash ell(v)\; =\; -\backslash lambda(iv)\; +\; i\backslash lambda(v)$$ giving the desired map.

Properties

The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps.

Anti-dual space

The vector space of all antilinear forms on a vector space

X is called the of X. If X is a topological vector space, then the vector space of all antilinear functionals on X, denoted by $\backslash overline^,$ is called the or simply the of X if no confusion can arise.

When

H is a normed space then the canonical norm on the (continuous) anti-dual space $\backslash overline^,$ denoted by $\backslash |f\backslash |\_,$ is defined by using this same equation: $$\backslash |f\backslash |\_\; ~:=~\; \backslash sup\_\; |f(x)|\; \backslash quad\; \backslash text\; f\; \backslash in\; \backslash overline^.$$ X ^{\prime} of X, which is defined by$$\backslash |f\backslash |\_\; ~:=~\; \backslash sup\_\; |f(x)|\; \backslash quad\; \backslash text\; f\; \backslash in\; X^.$$

Canonical isometry between the dual and anti-dual

\overline{f} of a functional f is defined by sending x \in \operatorname{domain} f to $\backslash overline.$ It satisfies $$\backslash |f\backslash |\_\; ~=~\; \backslash left\backslash |\backslash overline\backslash right\backslash |\_\; \backslash quad\; \backslash text\; \backslash quad\; \backslash left\backslash |\backslash overline\backslash right\backslash |\_\; ~=~\; \backslash |g\backslash |\_$$for every f \in X ^{\prime} and every $g\; \backslash in\; \backslash overline^.$ This says exactly that the canonical antilinear bijection defined by$$\backslash operatorname\; ~:~\; X^\; \backslash to\; \backslash overline^\; \backslash quad\; \backslash text\; \backslash quad\; \backslash operatorname(f)\; :=\; \backslash overline$$ as well as its inverse \operatorname{Cong} ^{-1} ~:~ \overline{X} ^{\prime} \to X ^{\prime} are antilinear isometries and consequently also homeomorphisms.

If

F = \R then X ^{\prime} = \overline{X} ^{\prime} and this canonical map \operatorname{Cong} : X ^{\prime} \to \overline{X} ^{\prime} reduces down to the identity map.

Inner product spaces

If

X is an inner product space then both the canonical norm on X ^{\prime} and on \overline{X} ^{\prime} satisfies the parallelogram law, which means that the polarization identity can be used to define a and also on \overline{X} ^{\prime} , which this article will denote by the notations $$\backslash langle\; f,\; g\; \backslash rangle\_\; :=\; \backslash langle\; g\; \backslash mid\; f\; \backslash rangle\_\; \backslash quad\; \backslash text\; \backslash quad\; \backslash langle\; f,\; g\; \backslash rangle\_\; :=\; \backslash langle\; g\; \backslash mid\; f\; \backslash rangle\_$$ where this inner product makes X ^{\prime} and \overline{X} ^{\prime} into Hilbert spaces. The inner products $\backslash langle\; f,\; g\; \backslash rangle\_$ and $\backslash langle\; f,\; g\; \backslash rangle\_$ are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by $f\; \backslash mapsto\; \backslash sqrt$) is consistent with the dual norm (that is, as defined above by the supremum over the unit ball); explicitly, this means that the following holds for every f \in X ^{\prime} : $$\backslash sup\_\; |f(x)|\; =\; \backslash |f\backslash |\_\; ~=~\; \backslash sqrt\; ~=~\; \backslash sqrt.$$

If

X is an inner product space then the inner products on the dual space X ^{\prime} and the anti-dual space $\backslash overline^,$ denoted respectively by $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\_$ and $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\_,$ are related by$$\backslash langle\; \backslash ,\backslash overline\backslash ,\; |\; \backslash ,\backslash overline\backslash ,\; \backslash rangle\_\; =\; \backslash overline\; =\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,f\backslash ,\; \backslash rangle\_\; \backslash qquad\; \backslash text\; f,\; g\; \backslash in\; X^$$and$$\backslash langle\; \backslash ,\backslash overline\backslash ,\; |\; \backslash ,\backslash overline\backslash ,\; \backslash rangle\_\; =\; \backslash overline\; =\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,f\backslash ,\; \backslash rangle\_\; \backslash qquad\; \backslash text\; f,\; g\; \backslash in\; \backslash overline^.$$

References

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. . (antilinear maps are discussed in section 3.3).
• Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. . (antilinear maps are discussed in section 4.6).

Notes and References

1. Book: Birkenhake, Christina. Complex Abelian Varieties . 2004 . Springer Berlin Heidelberg. Herbert Lange . 978-3-662-06307-1. Second, augmented. Berlin, Heidelberg. 851380558.