����������� ������ 54 (2002), 1–7

UDK 517.984.4

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MEASURES OF NON-STRICT-SINGULARITY AND NON-STRICT-COSINGULARITY

Snežana Živković

Abstract. In this paper we investigate a new measure of non-strict-singularity and a new measure of non-strict-cosingularity. Measures of non-strict-singularity and of non-strict-cosingularity have been investigated in [11], [8], [12], [7], [9], [15].

1. Introduction and preliminaries

In this paper X, Y and Z are complex Banach spaces, B(X, Y) (K(X, Y)) the set of all bounded (compact) linear operators from X into Y. We shall write B(X) (K(X)) instead of B(X, X) (K(X, X)).

An operator T ∈ B(X, Y) is in Φ_{+}(X, Y) (Φ_{−}(X, Y)) if the range R(T) is closed in Y and the dimension α(T) of the null space N(T) of T is finite (the codimension β(T) of R(T) in Y is finite). Operators in Φ_{+}(X, Y) ∪ Φ_{−}(X, Y) are called semi-Fredholm operators. We set Φ(X, Y) = Φ_{+}(X, Y) ∩ Φ_{−}(X, Y). The operators in Φ(X, Y) are called Fredholm operators. We shall write Φ_{+}(X) (resp. Φ_{−}(X), Φ(X)) instead of Φ_{+}(X, X) (resp. Φ_{−}(X, X), Φ(X)).

Let B_{X} denote the closed unit ball of X. Let T ∈ B(X, Y) and

m(T) = inf{ ∥T x∥ : ∥x∥ = 1 }

be the minimum modulus of T, and let

q(T) = sup{ ε ⩾ 0 : εB_{Y} ⊂ T B_{X} }

be the surjection modulus of T.

If M is a subspace of X, then J_{M} will denote the embedding map of M into X, and if V is a subspace of Y, then Q_{V} will denote the canonical map of Y onto the quotient space Y/V.

An operator T ∈ B(X, Y) is strictly singular (T ∈ S(X, Y)) if, for every infinite dimensional (closed) subspace M of X, the restriction of T to M, T |_{M}, is not a homeomorphism, i.e., m(T |_{M}) = 0. An operator T ∈ B(X, Y) is strictly cosingular (T ∈ SC(X, Y)) if, for every infinite codimensional closed subspace V of Y the composition Q_{V} T is not surjective. It is well known that

AMS Subject Classification: 47 A 10, 47 A 53

Keywords and phrases: Strictly (co)singular operators, measures of non-strict-(co)singularity, semi-Fredholm operators.

K(X, Y) ⊂ S(X, Y) and K(X, Y) ⊂ SC(X, Y).  (1.1)

If Ω is a non-empty bounded subset of X, then the Hausdorff measure of noncompactness of Ω is denoted by χ(Ω), and defined as follows

χ(Ω) = inf{ ε > 0 : Ω has a finite ε-net in X }.

For A ∈ B(X, Y) the Hausdorff measure of noncompactness of A, denoted by ∥A∥_{χ}, is defined by

∥A∥_{χ} = inf{ k ⩾ 0 : χ_{Y} (AΩ) ⩽ k_{χ_{X}} (Ω), Ω ⊂ X is bounded }.

Recall that ([2])

∥A∥_{χ} = inf{ ∥Q_{V} A∥ : V is a subspace of Y, dim V < ∞ }.
For A ∈ B(X, Y), set (see [6])
∥A∥_{μ} = inf{ ∥AJ_{L}∥ : L closed subspace of X, codim L < ∞ }.

Recall that

∥A∥_{χ} = 0 ⟺ ∥A∥_{μ} = 0 ⟺ A ∈ K(X, Y).  (1.2)

For A ∈ B(X, Y), set

G_{M}(A) = inf_{N⊂M} ∥AJ_{N}∥, G(A) = G_{X}(A),
Δ_{M}(A) = sup_{N⊂M} G_{N}(A), Δ(A) = Δ_{X}(A),

where M, N denote closed infinite dimensional subspaces of X (see [11]). Δ is a measure of non-strict-singularity of operators, i.e.,

Δ(A) = 0 ⟺ A ∈ S(X, Y).  (1.3)

Weis [8] introduced for A ∈ B(X, Y) the following functions

K_{V} (A) = inf_{W ⊃V} ∥Q_{W} A∥, K(A) = K_{{0}}(A),
∇_{V} (A) = sup_{W ⊃V} K_{W} (A), ∇(A) = ∇_{{0}}(A),

where V, W denote closed infinite codimensional subspaces of Y.

∇ is a measure of non-strict-cosingularity, i.e.,

∇(A) = 0 ⟺ A ∈ SC(X, Y).  (1.4)

Recall that

∇(A + T) = ∇(A) for all T ∈ SC(X, Y),  (1.5)

and

K_{V} (A) = inf_{W ⊃V} ∥Q_{W} A∥_{χ},  (1.6)

where V, W denote closed infinite codimensional subspaces of Y (see [12, Summary and discussion, Remark 2] or [7, Example 5.3] or [15, Lemma 2.21).

Recall that ([11], [13])

\begin{aligned}G(A) > 0 & ⟺ A ∈ Φ_{+}(X, Y), \\ K(A) > 0 & ⟺ A ∈ Φ_{−}(X, Y). \end{aligned}  (1.7)

2. Results

Schechter [11] proved the next theorem.

Theorem 2.1. A ∈ Φ_{+}(X, Y) if and only if for each Banach space Z there is a constant c, 0 < c < ∞, such that

Δ(T) ⩽ cΔ(AT), T ∈ B(Z, X).

We can prove the dual theorem.

Theorem 2.2. A ∈ Φ_{−}(X, Y) if and only if for each Banach space Z there is a constant c, 0 < c < ∞, such that

∇(T) ⩽ c∇(T A), T ∈ B(Y, Z).  (2.2.1)

Proof. Let A ∈ Φ_{−}(X, Y). By [6, Theorem 5.5 and Theorem 3.1] it follows that there is a constant c, 0 < c < ∞, such that for each Banach space Z

∥T ∥_{χ} ⩽ c∥T A∥_{χ}, T ∈ B(Y, Z).  (2.2.2)

Let V be a closed subspace of Z with codim V = ∞ and ε > 0. From (1.6) it follows that there is a closed subspace W of Z such that W ⊃ V, codim W = ∞ and

∥Q_{W} T A∥_{χ} < K_{V} (T A) + ε.  (2.2.3)

From (1.6), (2.2.2) and (2.2.3) it follows that

K_{V} (T) ⩽ ∥Q_{W} T ∥_{χ} ⩽ c∥Q_{W} T A∥_{χ} ⩽ c(K_{V} (T A) + ε)
⩽ c(∇(T A) + ε).

Hence ∇(T) ⩽ c(∇(T A) + ε).

Assume A ∉ Φ_{−}(X, Y). By [1, Theorem 4.4.10] it follows that there is an operator C ∈ K(X, Y) such that codim \overline{R(A − C)} = ∞. Let V = \overline{R(A − C)}. Since Q_{V} (A−C) = 0, from (1.1) and (1.5) we get ∇(Q_{V} A) = ∇(Q_{V} (A−C)) = 0. Let M and N be closed subspaces of Y/V with codim M = ∞, N ⊃ M and codim N = ∞. Since ∥Q_{N}Q_{V} ∥ = 1 we get

∇(Q_{V}) = sup_{\begin{matrix}M⊂Y/V \\ codim M=∞ \end{matrix}} inf_{\begin{matrix}N⊃M \\ codim N=∞ \end{matrix}} ∥Q_{N}Q_{V} ∥ = 1.

Thus, there is no constant c, 0 < c < ∞, such that (2.2.1) holds.

Let S be a subset of a Banach space A. The perturbation class associated with S is denoted by P (S) and

P (S) = { a ∈ A : a + s ∈ S for all s ∈ S }.

The perturbation class associated with Φ_{+}(X, Y) (Φ_{−}(X, Y)) is denoted by P (Φ_{+}(X, Y)) (P (Φ_{−}(X, Y))).

For T ∈ B(X, Y), set (see [10], [14])

n_{P Φ_{+}} = ∥T ∥_{P Φ_{+}} = inf{ ∥T − P ∥ : P ∈ P (Φ_{+}(X, Y)) },
n_{P Φ_{−}} = ∥T ∥_{P Φ_{−}} = inf{ ∥T − P ∥ : P ∈ P (Φ_{−}(X, Y)) },

The next theorem is inspired by [3, Example 1].

Theorem 2.3. Let T ∈ B(X, Y). Then

m(T) ⩽ ∥T ∥_{P Φ_{+}} ⩽ ∥T ∥,  (2.3.1)
q(T) ⩽ ∥T ∥_{P Φ_{−}} ⩽ ∥T ∥.  (2.3.2)

Proof. (2.3.1) Assume P ∈ P (Φ_{+}(X, Y)). It implies P ∉ Φ_{+}(X, Y). By [1, Theorem 4.4.7] it follows that there is K ∈ K(X, Y) such that dim N(P − K) = ∞. Set M = N(P − K) and ε > 0. By (1.2) we get ∥P J_{M}∥_{μ} = ∥KJ_{M}∥_{μ} = 0. Hence there is a closed subspace V ⊂ M such that dim M/V < ∞ and ∥P J_{V} ∥ < ε. For x ∈ V we have

∥T x − P x∥ ⩾ ∥T x∥ − ∥P x∥ ⩾ m(T)∥x∥ − ε∥x∥.

It implies ∥T − P ∥ ⩾ ∥(T − P)J_{V} ∥ ⩾ m(T) − ε. Hence ∥T − P ∥ ⩾ m(T). Thus ∥T ∥_{P Φ_{+}} ⩾ m(T).

(2.3.2) Let P ∈ P (Φ_{−}(X, Y)). Then P ∉ Φ_{−}(X, Y). From [1, Theorem 4.4.10] it follows that there is K ∈ K(X, Y) such that codim \overline{R(P − K)} = ∞. Set U = \overline{R(P − K)}. From Q_{U}(P − K) = 0 and (1.2) it follows ∥Q_{U}P ∥_{χ} = ∥Q_{U}K∥_{χ} = 0. Hence for ε > 0 there is a finite dimensional subspace W ⊂ Y/U such that ∥Q_{W} Q_{U}P ∥ < ε. There is a closed subspace V ⊂ Y such that V ⊃ U and W = V/U. It is not difficult to see that the operator A: (Y/U)/(V/U) → Y/V defined by

A((y + U) + V/U) = y + V, y ∈ Y,

is an isometric isomorphism and AQ_{V/U}Q_{U} = Q_{V}. Hence ∥Q_{V} P ∥ = ∥Q_{V/U}Q_{U}P ∥. It follows that ∥Q_{V} P ∥ < ε. Hence

∥T − P ∥ ⩾ ∥Q_{V} (T − P)∥ ⩾ ∥Q_{V} T ∥ − ∥Q_{V} P ∥ ⩾ q(Q_{V} T) − ε ⩾ q(T) − ε.

Thus ∥T − P ∥ ⩾ q(T), and ∥T ∥_{P Φ_{−}} ⩾ q(T).

Measures of non-strict-singularity and non-strict-cosingularity 5

Now we use the notation of [7]: let, for T ∈ B(X, Y),

sn_{P Φ_{+}}(T) = sup_{M} n_{P Φ_{+}}(T J_{M}),
isn_{P Φ_{+}}(T) = inf_{M} sn_{P Φ_{+}}(T J_{M}),

where M denotes a closed infinite dimensional subspace of X and

sn′_{P Φ_{−}}(T) = sup_{U} n_{P Φ_{−}}(Q_{U}T),
isn′_{P Φ_{−}}(T) = inf_{U} sn′_{P Φ_{−}}(Q_{U}T),

where U denotes a closed infinite codimensional subspace of Y.

Zemánek [13] considered the following functions

u(A) = sup{ m(AJ_{W}) : W is a closed subspace of X with dim W = ∞ },
v(A) = sup{ q(Q_{V} A) : V is a closed subspace of Y with codim V = ∞ }.

From the definition of the strictly singular and strictly cosingular operators it is obvious that

\begin{aligned}u(A) = 0 & ⟺ A ∈ S(X, Y), \\ v(A) = 0 & ⟺ A ∈ SC(X, Y). \end{aligned}  (2.4)

For A ∈ B(X, Y) set (see [4], [5])

G_{u}(A) = inf{ u(AJ_{M}) : M is a closed subspace of X, dim M = ∞ },
K_{v}(A) = inf{ v(Q_{U}A) : U is a closed subspace of Y, codim U = ∞ }.

Recall that

\begin{aligned}G_{u}(A) > 0 & ⟺ A ∈ Φ_{+}(X, Y), \\ K_{v}(A) > 0 & ⟺ A ∈ Φ_{−}(X, Y). \end{aligned}  (2.5)

From (2.3.1) and (2.3.2) it follows

G_{u}(T) ⩽ isn_{P Φ_{+}}(T) ⩽ G(T),
K_{v}(T) ⩽ isn′_{P Φ_{−}}(T) ⩽ K(T). (2.6)

By (2.6), (1.7) and (2.5) we get

isn_{P Φ_{+}}(T) > 0 ⟺ T ∈ Φ_{+}(X, Y),
isn′_{P Φ_{−}}(T) > 0 ⟺ T ∈ Φ_{−}(X, Y). (2.7)

(2.7) follows also from [7, Theorem 2.3(2) and Theorem 3.3(2)].

For T ∈ B(X, Y), set

Δ_{P Φ_{+}}(T) = sup_{M} inf_{N⊂M} ∥T J_{N}∥_{P Φ_{+}},
∇_{P Φ_{−}}(T) = sup_{V} inf_{W ⊃V} ∥Q_{W} T ∥_{P Φ_{−}},

where M, N denote closed infinite dimensional subspaces of X and V, W denote closed infinite codimensional subspaces of Y.

Analogously as in [11] it can be proved that Δ_{P Φ_{+}} (∇_{P Φ_{−}}) is a seminorm. From (2.3.1) and (2.3.2) it follows

\begin{matrix}u & ⩽ Δ_{P Φ_{+}} ⩽ Δ, \\ v & ⩽ ∇_{P Φ_{−}} ⩽ ∇. \end{matrix}  (2.8)

By (2.8), (1.3), (1.4) and (2.4) we get that Δ_{P Φ_{+}} is a measure of non-strict-singularity and ∇_{P Φ_{−}} is a measure of non-strict-cosingularity, i.e.,

Δ_{P Φ_{+}}(T) = 0 ⟺ T ∈ S(X, Y),  (2.9)
∇_{P Φ_{−}}(T) = 0 ⟺ T ∈ SC(X, Y).  (2.10)

(2.9) and (2.10) follow also from [7, Theorem 2.4(2) and Theorem 3.3(2)]. It is well known that

S(X, Y) ⊂ P (Φ_{+}(X, Y)) and SC(X, Y) ⊂ P (Φ_{−}(X, Y)).

Theorem 2.4. Let X and Y be Banach spaces. Then: (2.11.1) S(X, Y) = P (Φ_{+}(X, Y)) if and only if from P ∈ P (Φ_{+}(X, Y)) it follows

P J_{M} ∈ P (Φ_{+}(M, Y)) for each closed infinite dimensional subspace M of X;

(2.11.2) SC(X, Y) = P (Φ_{−}(X, Y)) if and only if from P ∈ P (Φ_{−}(X, Y)) it follows Q_{V} P ∈ P (Φ_{−}(X, Y/V)) for each closed infinite codimensional subspace V of Y.

Proof. (2.11.1). Let S(X, Y) = P (Φ_{+}(X, Y)). Suppose M is a closed infinite dimensional subspace of X and P ∈ P (Φ_{+}(X, Y)). Then P ∈ S(X, Y). It implies P J_{M} ∈ S(M, Y) ⊂ P (Φ_{+}(M, Y)).

Assume that for each closed infinite dimensional subspace M of X from P ∈ P (Φ_{+}(X, Y)) it follows P J_{M} ∈ P (Φ_{+}(M, Y)). Hence for T ∈ B(X, Y) we get

∥T J_{M}∥_{P Φ_{+}} = inf{ ∥T J_{M} − P_{1}∥ : P_{1} ∈ P (Φ_{+}(M, Y)) }
⩽ inf{ ∥T J_{M} − P J_{M}∥ : P ∈ P (Φ_{+}(X, Y)) }
⩽ inf{ ∥T − P ∥ : P ∈ P (Φ_{+}(X, Y)) } ⩽ ∥T ∥_{P Φ_{+}}.

Therefore

Δ_{P Φ_{+}}(T) ⩽ ∥T ∥_{P Φ_{+}}.  (2.11.3)

If T ∈ P (Φ_{+}(X, Y)), then ∥T ∥_{P Φ_{+}} = 0. By (2.11.3) it follows that Δ_{P Φ_{+}}(T) = 0. From (2.9) we get T ∈ S(X, Y). Thus S(X, Y) = P (Φ_{+}(X, Y)).

(2.11.2). Analogously to (2.11.1).

Acknowledgement. I am grateful to Dragan -Dor - dević for helpful conversations.

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(received 03.09.1998, in revised form 15.09.2000)

University of Niš, Faculty of Sciences, Department of Mathematics, Višegradska 33, 18000 Niš, Yugoslavia E-mail: zana@pmf.pmf.ni.ac.yu