In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a branch of
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 ...
, a strictly singular operator is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
between normed spaces which is not bounded below on any infinite-dimensional subspace.
Definitions.
Let ''X'' and ''Y'' be
normed linear spaces, and denote by ''B(X,Y)'' the space of
bounded operators
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
of the form
. Let
be any subset. We say that ''T'' is bounded below on
whenever there is a constant
such that for all
, the inequality
holds. If ''A=X'', we say simply that ''T'' is bounded below.
Now suppose ''X'' and ''Y'' are Banach spaces, and let
and
denote the respective identity operators. An operator
is called inessential whenever
is a
Fredholm operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X ...
for every
. Equivalently, ''T'' is inessential if and only if
is Fredholm for every
. Denote by
the set of all inessential operators in
.
An operator
is called strictly singular whenever it fails to be bounded below on any infinite-dimensional subspace of ''X''. Denote by
the set of all strictly singular operators in
. We say that
is finitely strictly singular whenever for each
there exists
such that for every subspace ''E'' of ''X'' satisfying
, there is
such that
. Denote by
the set of all finitely strictly singular operators in
.
Let
denote the closed unit ball in ''X''. An operator
is compact whenever
is a relatively norm-compact subset of ''Y'', and denote by
the set of all such compact operators.
Properties.
Strictly singular operators can be viewed as a generalization of
compact operators, as every compact operator is strictly singular. These two classes share some important properties. For example, if ''X'' is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
and ''T'' is a strictly singular operator in ''B(X)'' then its
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
satisfies the following properties: (i) the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of
is at most countable; (ii)
(except possibly in the trivial case where ''X'' is finite-dimensional); (iii) zero is the only possible
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
of
; and (iv) every nonzero
is an eigenvalue. This same "spectral theorem" consisting of (i)-(iv) is satisfied for inessential operators in ''B(X)''.
Classes
,
,
, and
all form norm-closed
operator ideals. This means, whenever ''X'' and ''Y'' are Banach spaces, the component spaces
,
,
, and
are each closed subspaces (in the operator norm) of ''B(X,Y)'', such that the classes are invariant under composition with arbitrary bounded linear operators.
In general, we have
, and each of the inclusions may or may not be strict, depending on the choices of ''X'' and ''Y''.
Examples.
Every bounded linear map
, for
,
, is strictly singular. Here,
and
are
sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
s. Similarly, every bounded linear map
and
, for
, is strictly singular. Here
is the Banach space of sequences converging to zero. This is a corollary of Pitt's theorem, which states that such ''T'', for ''q'' < ''p'', are compact.
If