MATEMATIQKI VESNIK

58 (2006), 1–6

UDK 517.984

originalni nauqni rad research paper

OPERATIONAL QUANTITIES DERIVED FROM THE MINIMUM MODULUS

Manuel Gonzáles^{†} and Antonio Martinón^{‡}

Abstract. The minimum modulus γ(T) of an operator T is useful in perturbation theory because it characterizes the operators with closed range. Here we study the operational quantities derived from γ(T). We show that the behavior of some of these quantities depends largely on whether the null space of T is finite dimensional or infinite dimensional.

1. Introduction

For every (linear bounded) non-zero operator T ∈ L(X, Y), where X and Y are Banach spaces, the minimum modulus is defined by

γ(T) := inf_{x∉N(T)} \frac{∥T x∥}{dist (x, N(T))}.

For T = 0 we set γ(0) = 0. It is well known that γ(T) > 0 if and only if T has closed range and T ≠ 0 [1, Theorem IV.1.6].

Here we study the operational quantities that can be derived from the minimum modulus γ(T).

In the preliminaries, we give a description of the procedure to derive the quantities associated to a given quantity. This procedure, applied to the norm n(T) ≡ ∥T ∥, provides three quantities in, sin and i^{∗}n which have been studied (with a different notation) in [9], [4], [7] and [2, 3]. Applied to the injection modulus j(T) := inf{ ∥T x∥ : x ∈ X, ∥x∥ = 1 }, it provides three quantities s^{∗}j, sj and isj which have been studied in [7] and [2, 3].

These operational quantities have been applied to characterize the classes of operators in Fredholm theory: see Theorem 1 in the preliminaries. For an excellent exposition of the Fredholm theory using operational quantities we refer to Chapter 14 in Schechter’s book [8].

AMS Subject Classification: 47 A 53

†Supported in part by DGI (Spain), Proyecto BFM2001-1147

‡Supported in part by Ministerio de Ciencia y Tecnologia (Spain) and FEDER, BFM2003-07139.

From the minimum modulus γ we can derive eight quantities isγ, sisγ, i^{∗}sγ, sγ, iγ, s^{∗}iγ, isiγ and siγ. In this paper we obtain that iγ agrees with the injection modulus j and we derive, consequently, that the quantities s^{∗}iγ, isiγ and siγ are the known quantities s^{∗}j, isj and sj, respectively.

On the other hand, we prove that if the operator T has finite dimensional null space N(T), then sj(T) ⩽ sγ(T) ⩽ 2sj(T) and we obtain that sγ, sisγ and i^{∗}sγ characterize the strictly singular operators, while isγ characterizes the upper semi-Fredholm operators. In the case N(T) infinite dimensional, we have sγ(T) = n(T) and we obtain that the quantities isγ, sisγ and i^{∗}sγ characterize the upper semi-Fredholm operators, the strictly singular operators and the compact operators, respectively.

Along the paper, X, Y, Z and W are infinite dimensional Banach spaces. By L(X, Y) we denote the space of all (linear continuous) operators from X into Y. For a (closed) subspace M of X, J_{M} is the inclusion of M into X. An operator T ∈ L(X, Y) is upper semi-Fredholm if its range is closed and its null space is finite dimensional; it is strictly singular if no restriction T J_{M} of T to a closed infinite dimensional subspace M of X is an isomorphism.

2. Preliminaries

Roughly speaking, an operational quantity is a procedure a which determines a real number a(T) ⩾ 0 for every operator T. For two quantities a and b we write a ⩽ b when

a(T) ⩽ b(T), for every operator T .

We say that the operational quantities a and b are equivalent if αa ⩽ b ⩽ βa, for some β > α > 0.

Given an operational quantity a and denoting by J_{M} the canonical inclusion of M into X, for every operator T ∈ L(X, Y), where X is an infinite dimensional space, we derive the following basic quantities:

s^{∗}a(T) := sup{a(T J_{P}) : P finite codimensional subspace of X},
sa(T) := sup{a(T J_{M}) : M infinite dimensional subspace of X},
i^{∗}a(T) := inf{a(T J_{P}) : P finite codimensional subspace of X},
ia(T) := inf{a(T J_{M}) : M infinite dimensional subspace of X}.

Repeating the procedure, we could derive new quantities like sia, sisa, i^{∗}issa, …, but surprisingly we obtain only three different new quantities when a is monotone [5]:

If a is increasing, in the sense that a(T J_{M}) ⩽ a(T) for every M, then ia, sia and i^{∗}a are the only new quantities, and they satisfy

ia ⩽ sia ⩽ i^{∗}a ⩽ a.

If a is decreasing, in the sense that a(T J_{M}) ⩾ a(T) for every M, then sa, isa and s^{∗}a are the only new quantities, and they satisfy

a ⩽ s^{∗}a ⩽ isa ⩽ sa.

The norm n(T) ≡ ∥T ∥ is an increasing quantity. So we get i^{∗}n, in and sin. The injection modulus j(T) is decreasing. So we get s^{∗}j, sj and isj. There are some relations between these quantities: isj ⩽ in and sj ⩽ sin.

The operational quantities associated to n and j have been applied to characterize the classes of operators in Fredholm theory:

Theorem 1. [2, 3], [7], [9]

1. i^{∗}n(T) = 0 ⇔ T compact,

2. sin(T) = 0 ⇔ sj(T) = 0 ⇔ T strictly singular,

3. in(T) > 0 ⇔ s^{∗}j(T) > 0 ⇔ isj(T) > 0 ⇔ T upper semi-Fredholm.

The quantity isj has been introduced in [2]; moreover, it was proved in [3] that, although

s^{∗}j ⩽ isj ⩽ in ,

these quantities are pairwise non-equivalent.

3. Main results

The operational quantity γ is not monotone, but iγ is decreasing and sγ is increasing. Hence we derive from γ the quantities

iγ ⩽ s^{∗}iγ ⩽ isiγ ⩽ siγ;
isγ ⩽ sisγ ⩽ i^{∗}sγ ⩽ sγ.

We begin by studying the operational quantities associated with iγ.

Theorem 2. For every T ∈ L(X, Y), iγ(T) = j(T).

Proof. Let T in L(X, Y). If N(T) = {0} or N(T) is infinite dimensional, then the statement is obvious. So we assume that 0 < dim N(T) < ∞, hence j(T) = 0.

We choose x ∈ N(T) with ∥x∥ = 1 and y ∈ X \ N(T), and we write

X = N(T) ⊕ ⟨y⟩ ⊕ M,

where ⟨y⟩ is the subspace generated by y and M is a closed complement of N(T) ⊕ ⟨y⟩. Moreover, denoting y_{n} = (1/n)y, we define

M_{n} := M ⊕ ⟨x + y_{n}⟩.
Suppose that z ∈ M_{n} ∩ N(T) then
z = m + λ(x + y_{n}) = λx + λy_{n} + m,

for some m ∈ M and some scalar λ. Thus λy_{n} = m = 0, and we conclude z = 0.

Since T J_{M_{n}} is injective, ∥x + y_{n}∥ → 1 and ∥T (x + y_{n})∥ = (1/n)∥T y∥ → 0, we have γ(T J_{M_{n}}) → 0; hence iγ(T) = 0.

Corollary 1. Let T ∈ L(X, Y).

1. iγ(T) > 0 ⇔ T isomorphism (into);

2. siγ(T) = 0 ⇔ T strictly singular;

3. isiγ(T) > 0 ⇔ s^{∗}iγ(T) > 0 ⇔ T upper semi-Fredholm.

Proof. From iγ = j we derive s^{∗}iγ = s^{∗}j, isiγ = isj and siγ = sj. From Theorem 1 we obtain the statement.

Now we study the quantities associated to sγ. We will see that the behavior of some of these quantities depends largely on whether the null space of T is finite dimensional or infinite dimensional.

Lemma 1. Let N be a finite dimensional subspace of X, and let 0 < ε < 1. For every infinite dimensional subspace M of X, there exists a finite codimensional subspace M_{ε} of M such that, for every x ∈ M_{ε},

∥x∥ ⩽ (2 + ε) dist(x, N).

Proof. Let {y_{1}, …, y_{k}} be an (ε/2)-net in the unit sphere of N. We choose f_{1}, …, f_{k} in the unit sphere of the dual space X^{∗} of X so that f_{i}(y_{i}) = 1 for i = 1, 2, …, k, and take

M_{ε} := {x ∈ M : f_{1}(x) = ⋯ = f_{n}(x) = 0}.

Let x ∈ M_{ε}. For each y ∈ N we denote z_{i} := ∥y∥y_{i} (1 ⩽ i ⩽ k). Then

∥y − x∥ ⩾ ∥x − z_{i}∥ − ∥y − z_{i}∥ ⩾ f_{i}(z_{i}) − (ε/2)∥y∥ = ∥y∥ − (ε/2)∥y∥ ⩾ \frac{∥y∥}{1 + ε}

for some i. Hence, ∥y∥ ⩽ (1 + ε)∥y − x∥ for each y in N. From this we obtain

∥x∥ ⩽ ∥x − y∥ + ∥y∥ ⩽ (2 + ε)∥x − y∥ ,

for each y in N, and this implies ∥x∥ ⩽ (2 + ε) dist(x, N).

Theorem 3. Let T ∈ L(X, Y). If N(T) is finite dimensional, then

sj(T) ⩽ sγ(T) ⩽ 2sj(T).

Proof. Let M be an infinite dimensional subspace of X, and let ε > 0. By Lemma 1, there exists a finite codimensional subspace M_{ε} of M such that for x ∈ M_{ε},

∥x∥ ⩽ (2 + ε) dist(x, N(T)) ⩽ (2 + ε) dist(x, N(T J_{M})).

Then

γ(T J_{M}) = inf_{x∈M,T x≠0} \frac{∥T x∥}{dist(x, N(T J_{M}))}

⩽ inf_{x∈M_{ε},T x≠0} \frac{∥T x∥}{dist(x, N(T J_{M}))}

⩽ inf_{x∈M_{ε},T x≠0} \frac{∥T x∥}{∥x∥}(2 + ε) = (2 + ε)j(T J_{M_{ε}}).

Hence, γ(T J_{M}) ⩽ (2 + ε)sj(T), so sγ(T) ⩽ 2sj(T).

The inequality sj(T) ⩽ sγ(T) is obvious.

Corollary 2. Let T ∈ L(X, Y). If N(T) is finite dimensional, then

1. isj(T) ⩽ isγ(T) ⩽ 2isj(T),

2. sj(T) ⩽ sisγ(T) ⩽ 2sj(T),

3. sj(T) ⩽ i^{∗}sγ(T) ⩽ 2sj(T).

Proof. Note that N(T J_{M}) is also finite dimensional for each M.

1. For every infinite dimensional subspace M of X we have that sj(T J_{M}) ⩽ sγ(T J_{M}) ⩽ 2sj(T J_{M}), hence isj(T) ⩽ isγ(T) ⩽ 2isj(T).

2. Similar to 1., taking into account sisj(T) = sj(T) [5].

3. Similar to 1., taking into account i^{∗}sj(T) = sj(T) [5]. The following result shows that the property isγ(T) > 0 is preserved by taking

products.

Corollary 3. Let T ∈ L(X, Y) and S ∈ L(Y, Z). If N(T) and N(S) are finite dimensional, then

isγ(S) isγ(T) ⩽ 4 isγ(ST)

Proof. It is known that isj(S) isj(T) ⩽ isj(ST) [2], hence isγ(S) isγ(T) ⩽ 2isj(S) 2isj(T) ⩽ 4isj(ST) ⩽ 4isγ(ST).

Theorem 4. For each operator T ∈ L(X, Y) with N(T) infinite dimensional, sγ(T) = n(T).

Proof. For T = 0 the result is obvious. Suppose T ≠ 0. For each x ∉ N(T) we put M_{x} := N(T) ⊕ ⟨x⟩, where ⟨x⟩ is the subspace generated by x.

For every y ∈ M_{x}, y = λx + z, z ∈ N(T), we obtain

\frac{∥T x∥}{∥x∥} = \frac{|λ| ∥T x∥}{|λ| ∥x∥} = \frac{∥T y∥}{∥y − z∥} ⩽ \frac{∥T y∥}{dist(y, N(T))}.

Thus n(T) ⩽ γ(T J_{M_{x}}) ⩽ sγ(T).

Note that for every T ≠ 0 with R(T) finite dimensional (hence N(T) is infinite dimensional), sj(T) = 0 ≠ n(T). Thus sj and sγ are not equivalent.

Corollary 4. Let T ∈ L(X, Y).

1. isγ(T) > 0 ⇔ T is upper semi-Fredholm.

2. If N(T) is finite dimensional, then

sγ(T) = 0 ⇔ sisγ(T) = 0 ⇔ i^{∗}sγ(T) = 0 ⇔ T is strictly singular.

3. If N(T) is infinite dimensional, then (a) sγ(T) = 0 ⇔ T = 0;

(b) sisγ(T) = 0 ⇔ T is strictly singular;

(c) i^{∗}sγ(T) = 0 ⇔ T is compact.

Proof. (1) [6, Example 5.1].

(2) It is immediate from Theorem 1, Theorem 3 and Corollary 2.

(3) (a) Theorem 4.

(b) sisγ(T) = 0 is equivalent to isγ(T J_{M}) = 0 for every infinite dimensional subspace M of X, which is equivalent by (1) to T J_{M} is not an upper semi-Fredholm operator, and consequently to T strictly singular.

(c) Since i^{∗}sγ(T) is the infimum of sγ(T J_{P}) = n(T J_{P}) where P runs over the finite codimensional subspaces of X, from Theorem 1, we obtain i^{∗}sγ(T) = i^{∗}n(T) = 0 if and only if T is compact.

REFERENCES

[1] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966 [Dover, New York, 1985].

[2] M. González, A. Martinón, Fredholm Theory and Space Ideals, Bol. U.M.I. (7) 7-B (1993), 473–488.

[3] M. González, A. Martinón, Operational quantities characterizing semi-Fredholm operators, Studia Math. 114 (1995), 13–27.

[4] A. Lebow, M. Schechter, Semigroups of operators and measures of noncompactness, J. Funct. Anal. 7 (1971), 1–26.

[5] A. Martinón, Generating real maps on a biordered set, Comment. Math. Univ. Carolinae 32 (1991), 265–272.

[6] A. Martinón, Operational quantities, Comment. Math. Univ. Carolinae 38 (1997), 471–484.

[7] M. Schechter, Quantities related to strictly singular operators, Indiana Univ. Math. J. 21 (1972), 1061–1071.

[8] M. Schechter, Principles of Functional Analysis, American Mathematical Society, Providence, Rhode Island, 2002.

[9] A. A. Sedaev, The structure of certain linear operators (in Russian), Mat. Issled. 5 (1970), 166–175.

(received 11.02.2003, in revised form 23.12.2004)

Manuel Gonzáles, Departamento de Matemáticas, Universidad de Cantabria, 39071 Santander, Spain

E-mail: gonzalem@unican.es

Antonio Martinón, Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain

E-mail: anmarce@ull.es