near operators

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1 Perturbations and small perturbations: definitions and some results

We start our discussion on the Campanato theory of near operators with some preliminary tools.

Let X,Y be two sets and let a metric d be defined on Y. If F:XY is an injective map, we can define a metric on X by putting


Indeed, dF is zero if and only if x=x′′ (since F is injectivePlanetmathPlanetmath); dF is obviously symmetricPlanetmathPlanetmathPlanetmathPlanetmath and the triangle inequalityMathworldMathworldPlanetmath follows from the triangle inequality of d.

Moreover, if F(X) is a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath subspaceMathworldPlanetmathPlanetmath of Y, then X is complete with respect to the metric dF.

Indeed, let (un) be a Cauchy sequenceMathworldPlanetmathPlanetmath in X. By definition of d, then (F(un)) is a Cauchy sequence in Y, and in particular in F(X), which is complete. Thus, there exists y0=F(x0)F(X) which is limit of the sequenceMathworldPlanetmathPlanetmath (F(un)). x0 is the limit of (xn) in (X,dF), which completes the proof.

A particular case of the previous statement is when F is onto (and thus a bijection) and (Y,d) is complete.

Similarly, if F(X) is compactPlanetmathPlanetmath in Y, then X is compact with the metric dF.

Definition 1.1

Let X be a set and Y be a metric space. Let F,G be two maps from X to Y. We say that G is a perturbation of F if there exist a constant k>0 such that for each x,x′′X one has:

Remark 1.2

In particular, if F is injective then G is a perturbation of F if G is uniformly continuousPlanetmathPlanetmath with respect to the metric induced on X by F.

Definition 1.3

In the same hypothesisMathworldPlanetmathPlanetmath as in the previous definition, we say that G is a small perturbation of F if it is a perturbation of constant k<1.

Theorem 1.4

Let X be a set and (Y,d) be a complete metric space. Let F,G be two mappings from X to Y such that:

  1. 1.

    F is bijectiveMathworldPlanetmath;

  2. 2.

    G is a small perturbation of F.

Then, there exists a unique uX such that G(u)=F(u)


The hypothesis (1) ensures that the metric space (X,dF) is complete. If we now consider the function T:XX defined by


we note that, by (2), we have


where k(0,1) is the constant of the small perturbation; note that, by the definition of dF and applying FF-1 to the first side, the last equation can be rewritten as


in other words, since k<1, T is a contractionPlanetmathPlanetmath in the complete metric space (X,dF); therefore (by the classical Banach-Caccioppoli fixed point theorem) T has a unique fixed pointPlanetmathPlanetmathPlanetmath: there exist uX such that T(u)=u; by definition of T this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to G(u)=F(u), and the proof is hence complete. ∎

Remark 1.5

The hypothesis of the theoremMathworldPlanetmath can be generalized as such: let X be a set and Y a metric space (not necessarily complete); let F,G be two mappings from X to Y such that F is injective, F(X) is complete and G(X)F(X); then there exists uX such that G(u)=F(u).

(Apply the theorem using F(X) instead of Y as target space.)

Remark 1.6

The Banach-Caccioppoli fixed point theorem is obtained when X=Y and F is the identity.

We can use theorem 1.4 to prove a result that applies to perturbations which are not necessarily small (i.e. for which the constant k can be greater than one). To prove it, we must assume some supplemental structureMathworldPlanetmath on the metric of Y: in particular, we have to assume that the metric d is invariantMathworldPlanetmath by dilations, that is that d(αy,αy′′)=αd(y,y′′) for each y,y′′Y. The most common case of such a metric is when the metric is deduced from a norm (i.e. when Y is a normed spaceMathworldPlanetmath, and in particular a Banach spaceMathworldPlanetmath). The result follows immediately:

Corollary 1.7

Let X be a set and (Y,d) be a complete metric space with a metric d invariant by dilations. Let F,G be two mappings from X to Y such that F is bijective and G is a perturbation of F, with constant K>0.

Then, for each M>K there exists a unique uMX such that G(u)=MF(u)


The proof is an immediate consequence of theorem 1.4 given that the map G~(u)=G(u)/M is a small perturbation of F (a property which is ensured by the dilation invariance of the metric d). ∎

We also have the following

Corollary 1.8

Let X be a set and (Y,d) be a complete, compact metric space with a metric d invariant by dilations. Let F,G be two mappings from X to Y such that F is bijective and G is a perturbation of F, with constant K>0.

Then there exists at least one uKX such that G(u)=KF(u)


Let (an) be a decreasing sequence of real numbers greater than one, converging to one (an1) and let Mn=anK for each n. We can apply corollary 1.7 to each Mn, obtaining a sequence un of elements of X for which one has

G(un)=MnF(un). (1)

Since (X,dF) is compact, there exist a subsequenceMathworldPlanetmath of un which convergesPlanetmathPlanetmath to some u; by continuity of G and F we can pass to the limit in (1), obtaining


which completes the proof. ∎

Remark 1.9

For theorem 1.8 we cannot ensure uniqueness of u, since in general the sequence un may change with the choice of an, and the limit might be different. So the corollary can only be applied as an existence theoremMathworldPlanetmath.

2 Near operators

We can now introduce the concept of near operators and discuss some of their properties.

A historical remark: Campanato initially introduced the concept in Hilbert spacesMathworldPlanetmath; subsequently, it was remarked that most of the theory could more generally be applied to Banach spaces; indeed, it was also proven that the basic definition can be generalized to make part of the theory available in the more general environment of metric vector spacesMathworldPlanetmath.

We will here discuss the theory in the case of Banach spaces, with only a couple of exceptions: to see some of the extra properties that are available in Hilbert spaces and to discuss a generalization of the Lax-Milgram theorem to metric vector spaces.

2.1 Basic definitions and properties

Definition 2.1

Let X be a set and Y a Banach space. Let A,B be two operators from X to Y. We say that A is near B if and only if there exist two constants α>0 and k(0,1) such that, for each x,x′′X one has


In other words, A is near B if B-αA is a small perturbation of B for an appropriate value of α.

Observe that in general the property is not symmetric: if A is near B, it is not necessarily true that B is near A; as we will briefly see, this can only be proven if α<1/2, or in the case that Y is a Hilbert space, by using an equivalent condition that will be discussed later on. Yet it is possible to define a topologyMathworldPlanetmathPlanetmath with some interesting properties on the space of operators, by using the concept of nearness to form a base.

The core point of the nearness between operators is that it allows us to “transfer” many important properties from B to A; in other words, if B satisfies certain properties, and A is near B, then A satisfies the same properties. To prove this, and to enumerate some of these “nearness-invariant” properties, we will emerge a few important facts.

In what follows, unless differently specified, we will always assume that X is a set, Y is a Banach space and A,B are two operators from X to Y.

Lemma 2.2

If A is near B then there exist two positivePlanetmathPlanetmath constants M1,M2 such that


We have:


and hence


which is the first inequalityMathworldPlanetmath with M1=α/(1-k) (which is positive since k<1).

But also


and hence


which is the second inequality with M2=(1+k)/α. ∎

The most important corollary of the previous lemma is the following

Corollary 2.3

If A is near B then two points of X have the same image under A if and only if the have the same image under B.

We can express the previous concept in the following formal way: for each y in B(X) there exist z in Y such that A(B-1(y))={z} and conversely. In yet other words: each fiber of A is a fiber (for a different point) of B, and conversely.

It is therefore possible to define a map TA:B(X)Y by putting TA(y)=z; the range of TA is A(X). Conversely, it is possible to define TB:A(X)Y, by putting TB(z)=y; the range of TB is B(X). Both maps are injective and, if restricted to their respective ranges, one is the inversePlanetmathPlanetmathPlanetmathPlanetmath of the other.

Also observe that TB and TA are continuousMathworldPlanetmathPlanetmath. This follows from the fact that for each xX one has


and that the lemma ensures that given a sequence (xn) in X, the sequence (B(xn)) converges to B(x0) if and only if (A(xn)) converges to A(x0).

We can now list some invariant properties of operators with respect to nearness. The properties are given in the form “if and only if” because each operator is near itself (therefore ensuring the “only if” part).

  1. 1.

    a map is injective if and only if it is near an injective operator;

  2. 2.

    a map is surjective if and only if it is near a surjective operator;

  3. 3.

    a map is open if and only if it is near an open map;

  4. 4.

    a map has dense range if and only if it is near a map with dense range.

To prove (2) it is necessary to use theorem 1.4.

Another important property that follows from the lemma is that if there exist yY such that A-1(y)B-1(y), then it is A-1(y)=B-1(y): intersecting fibers are equal. (Campanato only stated this property for the case y=0 and called it “‘the kernel property”; I prefer to call it the “fiber persistence” property.)

2.1.1 A topology based on nearness

In this sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath we will show that the concept of nearness between operator can indeed be connected to a topological understanding of the set of maps from X to Y.

Let be the set of maps between X and Y. For each F and for each k(0,1) we let Uk(F) the set of all maps G such that F-G is a small perturbation of F with constant k. In other words, GUk(F) if and only if G is near F with constants 1,k.

The set 𝒰(F)={Uk(F)0<k<1} satisfies the axioms of the set of fundamental neighbourhoods. Indeed:

  1. 1.

    F belongs to each Uk(F);

  2. 2.

    Uk(F)Uh(F) if and only if k<h, and thus the intersectionMathworldPlanetmathPlanetmath property of neighbourhoods is trivial;

  3. 3.

    for each Uk(F) there exist Uh(F) such that for each GUh(F) there exist Uj(G)Uk(F).

This last property (permanence of neighbourhoods) is somewhat less trivial, so we shall now prove it.


Let Uk(F) be given.

Let Uh(F) be another arbitrary neighbourhood of F and let G be an arbitrary element in it. We then have:

F(x)-F(x′′)-(G(x)-G(x′′))hF(x)-F(x′′). (2)

but also (lemma 2.2)

(G(x)-G(x′′))(1+h)F(x)-F(x′′). (3)

Let also Uj(G) be an arbitrary neighbourhood of G and H an arbitrary element in it. We then have:

G(x)-G(x′′)-(H(x)-H(x′′))jG(x)-G(x′′). (4)

The nearness between F and H is calculated as such:

F(x)-F(x′′)-(H(x)-H(x′′))F(x)-F(x′′)-(G(x)-G(x′′))+G(x)-G(x′′)-(H(x)-H(x′′))hF(x)-F(x′′)+jG(x)-G(x′′)(h+j(1+h))F(x)-F(x′′). (5)

We then want h+j(1+h)k, that is j(k-h)/(1+h); the condition 0<j<1 is always satisfied on the right side, and the left side gives us h<k. ∎

It is important to observe that the topology generated this way is not a Hausdorff topology: indeed, it is not possible to separate F and F+y (where F and y is a constant element of Y). On the other hand, the subset of all maps with with a fixed valued at a fixed point (F(x0)=y0) is a Hausdorff subspace.

Another important characteristic of the topology is that the set of invertible operators from X to Y is open in (because a map is invertible if and only if it is near an invertible map). This is not true in the topology of uniform convergence, as is easily seen by choosing X=Y= and the sequence with genericPlanetmathPlanetmathPlanetmath element Fn(x)=x3-x/n: the sequence converges (in the uniform convergence topology) to F(x)=x3, which is invertible, but none of the Fn is invertible. Hence F is an element of which is not inside , and is not open.

2.2 Some applications

As we mentioned in the introduction, the Campanato theory of near operators allows us to generalize some important theorems; we will now present some generalizations of the Lax-Milgram theorem, and a generalization of the Riesz representation theorem.


Title near operators
Canonical name NearOperators
Date of creation 2013-03-22 14:03:21
Last modified on 2013-03-22 14:03:21
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 13
Author mathcam (2727)
Entry type Topic
Classification msc 54E40
Synonym Campanato theory of near operators
Defines perturbation
Defines small perturbation
Defines near operator