Strictly singular operator

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. 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 → Y {displaystyle T:X o Y} . Let A ⊆ X {displaystyle Asubseteq X} be any subset. We say that T is bounded below on A {displaystyle A} whenever there is a constant c ∈ ( 0 , ∞ ) {displaystyle cin (0,infty )} such that for all x ∈ A {displaystyle xin A} , the inequality ‖ T x ‖ ≥ c ‖ x ‖ {displaystyle |Tx|geq c|x|} holds. If A=X, we say simply that T is bounded below. Now suppose X and Y are Banach spaces, and let I d X ∈ B ( X ) {displaystyle Id_{X}in B(X)} and I d Y ∈ B ( Y ) {displaystyle Id_{Y}in B(Y)} denote the respective identity operators. An operator T ∈ B ( X , Y ) {displaystyle Tin B(X,Y)} is called inessential whenever I d X − S T {displaystyle Id_{X}-ST} is a Fredholm operator for every S ∈ B ( Y , X ) {displaystyle Sin B(Y,X)} . Equivalently, T is inessential if and only if I d Y − T S {displaystyle Id_{Y}-TS} is Fredholm for every S ∈ B ( Y , X ) {displaystyle Sin B(Y,X)} . Denote by E ( X , Y ) {displaystyle {mathcal {E}}(X,Y)} the set of all inessential operators in B ( X , Y ) {displaystyle B(X,Y)} . An operator T ∈ B ( X , Y ) {displaystyle Tin B(X,Y)} is called strictly singular whenever it fails to be bounded below on any infinite-dimensional subspace of X. Denote by S S ( X , Y ) {displaystyle {mathcal {SS}}(X,Y)} the set of all strictly singular operators in B ( X , Y ) {displaystyle B(X,Y)} . We say that T ∈ B ( X , Y ) {displaystyle Tin B(X,Y)} is finitely strictly singular whenever for each ϵ > 0 {displaystyle epsilon >0} there exists n ∈ N {displaystyle nin mathbb {N} } such that for every subspace E of X satisfying dim ( E ) ≥ n {displaystyle { ext{dim}}(E)geq n} , there is x ∈ E {displaystyle xin E} such that ‖ T x ‖ < ϵ ‖ x ‖ {displaystyle |Tx|<epsilon |x|} . Denote by F S S ( X , Y ) {displaystyle {mathcal {FSS}}(X,Y)} the set of all finitely strictly singular operators in B ( X , Y ) {displaystyle B(X,Y)} . Let B X = { x ∈ X : ‖ x ‖ ≤ 1 } {displaystyle B_{X}={xin X:|x|leq 1}} denote the closed unit ball in X. An operator T ∈ B ( X , Y ) {displaystyle Tin B(X,Y)} is compact whenever T B X = { T x : x ∈ B X } {displaystyle TB_{X}={Tx:xin B_{X}}} is a relatively norm-compact subset of Y, and denote by K ( X , Y ) {displaystyle {mathcal {K}}(X,Y)} the set of all such compact operators. 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 and T is a strictly singular operator in B(X) then its spectrum σ ( T ) {displaystyle sigma (T)} satisfies the following properties: (i) the cardinality of σ ( T ) {displaystyle sigma (T)} is at most countable; (ii) 0 ∈ σ ( T ) {displaystyle 0in sigma (T)} (except possibly in the trivial case where X is finite-dimensional); (iii) zero is the only possible limit point of σ ( T ) {displaystyle sigma (T)} ; and (iv) every nonzero λ ∈ σ ( T ) {displaystyle lambda in sigma (T)} is an eigenvalue. This same 'spectral theorem' consisting of (i)-(iv) is satisfied for inessential operators in B(X). Classes K {displaystyle {mathcal {K}}} , F S S {displaystyle {mathcal {FSS}}} , S S {displaystyle {mathcal {SS}}} , and E {displaystyle {mathcal {E}}} all form norm-closed operator ideals. This means, whenever X and Y are Banach spaces, the component spaces K ( X , Y ) {displaystyle {mathcal {K}}(X,Y)} , F S S ( X , Y ) {displaystyle {mathcal {FSS}}(X,Y)} , S S ( X , Y ) {displaystyle {mathcal {SS}}(X,Y)} , and E ( X , Y ) {displaystyle {mathcal {E}}(X,Y)} 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 K ( X , Y ) ⊂ F S S ( X , Y ) ⊂ S S ( X , Y ) ⊂ E ( X , Y ) {displaystyle {mathcal {K}}(X,Y)subset {mathcal {FSS}}(X,Y)subset {mathcal {SS}}(X,Y)subset {mathcal {E}}(X,Y)} , and each of the inclusions may or may not be strict, depending on the choices of X and Y. Every bounded linear map T : ℓ p → ℓ q {displaystyle T:ell _{p} o ell _{q}} , for 1 ≤ q , p < ∞ {displaystyle 1leq q,p<infty } , p ≠ q {displaystyle p eq q} , is strictly singular. Here, ℓ p {displaystyle ell _{p}} and ℓ q {displaystyle ell _{q}} are sequence spaces. Similarly, every bounded linear map T : c 0 → ℓ p {displaystyle T:c_{0} o ell _{p}} and T : ℓ p → c 0 {displaystyle T:ell _{p} o c_{0}} , for 1 ≤ p < ∞ {displaystyle 1leq p<infty } , is strictly singular. Here c 0 {displaystyle c_{0}} 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.