In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator between normed spaces which is not bounded below on any infinite-dimensional subspace.
Let X and Y be normed linear spaces, and denote by B(X,Y) the space of bounded operators of the form
T:X\toY
A\subseteqX
A
c\in(0,infty)
x\inA
\|Tx\|\geqc\|x\|
Now suppose X and Y are Banach spaces, and let
IdX\inB(X)
IdY\inB(Y)
T\inB(X,Y)
IdX-ST
S\inB(Y,X)
IdY-TS
S\inB(Y,X)
l{E}(X,Y)
B(X,Y)
An operator
T\inB(X,Y)
l{SS}(X,Y)
B(X,Y)
T\inB(X,Y)
\epsilon>0
n\inN
dim(E)\geqn
x\inE
\|Tx\|<\epsilon\|x\|
l{FSS}(X,Y)
B(X,Y)
Let
BX=\{x\inX:\|x\|\leq1\}
T\inB(X,Y)
TBX=\{Tx:x\inBX\}
l{K}(X,Y)
\sigma(T)
\sigma(T)
0\in\sigma(T)
\sigma(T)
λ\in\sigma(T)
Classes
l{K}
l{FSS}
l{SS}
l{E}
l{K}(X,Y)
l{FSS}(X,Y)
l{SS}(X,Y)
l{E}(X,Y)
In general, we have
l{K}(X,Y)\subsetl{FSS}(X,Y)\subsetl{SS}(X,Y)\subsetl{E}(X,Y)
Every bounded linear map
T:\ellp\to\ellq
1\leq,p<infty
p\neq
\ellp
\ellq
T:c0\to\ellp
T:\ellp\toc0
1\lep<infty
c0
If
1\leqp<q<infty
Ip,q\inB(\ellp,\ellq)
1<p<q<infty
B(\ellp,\ellq)
n | |
\ell | |
2 |
n\inN
l{K}(\ellp,\ellq)\subsetneql{FSS}(\ellp,\ellq)\subsetneql{SS}(\ellp,\ellq)
\ellq
l{SS}(\ellp,\ellq)=l{E}(\ellp,\ellq)
T\inB(X,\ellinfty)
l{K}(\ellp,\ellinfty)\subsetneql{FSS}(\ellp,\ellinfty)\subsetneql{SS}(\ellp,\ellinfty)\subsetneql{E}(\ellp,\ellinfty)
1<p<infty
The compact operators form a symmetric ideal, which means
T\inl{K}(X,Y)
T*\inl{K}(Y*,X*)
l{FSS}
l{SS}
l{E}
QZ:Y\toY/Z
y\mapstoy+Z
T\inB(X,Y)
QZT
l{SCS}(X,Y)
Theorem 1. Let X and Y be Banach spaces, and let
T\inB(X,Y)
Note that there are examples of strictly singular operators whose adjoints are neither strictly singular nor strictly cosingular (see Plichko, 2004). Similarly, there are strictly cosingular operators whose adjoints are not strictly singular, e.g. the inclusion map
I:c0\to\ellinfty
l{SS}
l{SCS}
Theorem 2. Let X and Y be Banach spaces, and let
T\inB(X,Y)
Aiena, Pietro, Fredholm and Local Spectral Theory, with Applications to Multipliers (2004), .
Plichko, Anatolij, "Superstrictly Singular and Superstrictly Cosingular Operators," North-Holland Mathematics Studies 197 (2004), pp239-255.