In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, there are many kinds of
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
involving
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s on
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. This article covers some important operator inequalities connected with
traces
Traces may refer to:
Literature
* ''Traces'' (book), a 1998 short-story collection by Stephen Baxter
* ''Traces'' series, a series of novels by Malcolm Rose
Music Albums
* ''Traces'' (Classics IV album) or the title song (see below), 1969
* ''Tra ...
of matrices.
[E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 ][B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).]
Basic definitions
Let H
''n'' denote the space of
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
× matrices, H
''n''+ denote the set consisting of
positive semi-definite × Hermitian matrices and H
''n''++ denote the set of
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be
trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the tra ...
and
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a sta ...
, in which case similar definitions apply, but we discuss only matrices, for simplicity.
For any real-valued function on an interval ⊂ ℝ, one may define a
matrix function
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.
This is used for defining the exponential of a matrix, which is involved in the ...
for any operator with
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
in by defining it on the eigenvalues and corresponding
projectors
A projector or image projector is an optical device that projects an image (or moving images) onto a surface, commonly a projection screen. Most projectors create an image by shining a light through a small transparent lens, but some newer types ...
as
:
given the
spectral decomposition
Operator monotone
A function defined on an interval ⊂ ℝ is said to be operator monotone if ∀, and all with eigenvalues in , the following holds,
:
where the inequality means that the operator is positive semi-definite. One may check that is, in fact, ''not'' operator monotone!
Operator convex
A function
is said to be operator convex if for all
and all with eigenvalues in , and
, the following holds
:
Note that the operator
has eigenvalues in
, since
and
have eigenvalues in .
A function
is operator concave if
is operator convex, i.e. the inequality above for
is reversed.
Joint convexity
A function
, defined on intervals
is said to be jointly convex if for all
and all
with eigenvalues in
and all
with eigenvalues in
, and any
the following holds
:
A function is jointly concave if − is jointly convex, i.e. the inequality above for is reversed.
Trace function
Given a function : ℝ → ℝ, the associated trace function on H
''n'' is given by
:
where has eigenvalues and Tr stands for a
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the operator.
Convexity and monotonicity of the trace function
Let : ℝ → ℝ be continuous, and let be any integer. Then, if
is monotone increasing, so
is
on H
''n''.
Likewise, if
is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, so is
on H
''n'', and
it is strictly convex if is strictly convex.
See proof and discussion in,
for example.
Löwner–Heinz theorem
For
, the function
is operator monotone and operator concave.
For
, the function
is operator monotone and operator concave.
For
, the function
is operator convex. Furthermore,
:
is operator concave and operator monotone, while
:
is operator convex.
The original proof of this theorem is due to
K. Löwner who gave a necessary and sufficient condition for to be operator monotone. An elementary proof of the theorem is discussed in
and a more general version of it in.
Klein's inequality
For all Hermitian × matrices and and all differentiable
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
s
: ℝ → ℝ with
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
, or for all positive-definite Hermitian × matrices and , and all differentiable
convex functions :(0,∞) → ℝ, the following inequality holds,
In either case, if is strictly convex, equality holds if and only if = .
A popular choice in applications is , see below.
Proof
Let
so that, for
,
:
,
varies from
to
.
Define
: