real linear operator theory and its applicationsmath.aalto.fi/~mhuhtane/realbeltrami_final.pdfreal...

Real linear operator theoryand its applications

Marko Huhtanen and Santtu Ruotsalainen

Abstract. Real linear operators arise in a range of applications of math-ematical physics. In this paper, basic properties of real linear operatorsare studied and their spectral theory is developed. Suitable extensionsof classical operator theoretic concepts are introduced. Providing a con-crete class, real linear multiplication operators are investigated and, mo-tivated by the Beltrami equation, related problems of unitary approxi-mation are addressed.

Mathematics Subject Classification (2010). Primary 47A10; Secondary47B38.

Keywords. Real linear operator on Hilbert space, spectrum, Beurlingtransform, multiplication operator.

1. Introduction

The present paper deals with the theory of bounded real linear operatorson a complex separable Hilbert space. The study of complex linear operatortheory is classical whereas a more intensive investigation of real linear opera-tors has only fairly recently started, mainly in various mathematical physicsapplications. Real linearity is central in studies related with problems in pla-nar elasticity [20], in the theory of quasiconformal mappings [15], and in theinverse problem of recovering electrical conductivity distribution in the plane[3]. In this paper basic concepts of bounded real linear operators are devel-oped. The spectrum is studied. The notions of compact and unitary operatorsare defined. The Beltrami operator is examined from the view-point of reallinear operator theory. Preceding real linear Toeplitz operator theory, reallinear multiplication operators are studied in detail.

To elucidate the role of real linear operators, consider the general Bel-trami differential equation in the complex plane C

∂f = ν∂f + µ∂f, (1.1)

Supported by the Academy of Finland.

2 M. Huhtanen and S. Ruotsalainen

where ∂ = (∂x + i∂y)/2, ∂ = (∂x − i∂y)/2, z = x+ iy, and µ, ν ∈ L∞(C). Itarises in the study of the two dimensional Calderon problem and in connec-tion with quasiconformal mappings; see [3], [15], [14], [2]. Under appropriateassumptions, solving Eq. (1.1) leads to examining the invertibility of the so-called Beltrami operator

I − (µ+ ντ)S (1.2)

on Lp(C)-spaces, where I denotes the identity operator, τ the complex con-jugation operator and S is the Beurling transform. In this way, the questionconcerning the invertibility of the Beltrami operator can be viewed as a reallinear spectral problem for the real linear operator (µ+ ντ)S.

The spectral problem for real linear operators is challenging in generalby the fact that the spectrum can be empty. For compact operators, it cancontain a continuum; for self-adjoint operators, it is not necessarily real. Thus,the classical complex linear classes of operators need to be carefully inspectedin the corresponding real linear case. Aside from these general investigations,we consider a real linear generalization of multiplication operators appearingin (1.2). We study their basic properties and expose their unitary approxi-mation. These investigations pave the way for real linear Toeplitz operatortheory.

The organization of the paper is as follows. Section 2 is concerned withthe general theory of bounded real linear operators and their spectral theoryin particular. Recent applications are described. Compact, unitary and finiterank operators are defined. In section 3, real linear multiplication operatorson function spaces are studied. Unitary and scaled unitary approximationof real linear operators of the form (µ + ντ)B, with B complex linear andunitary, is examined.

2. Real linear operators and their spectral theory

2.1. Basic definitions

Let H be a complex separable Hilbert space. An operator A on H is said tobe real linear if

A(x+ y) = Ax+Ay and A(λx) = λAx

for all x, y ∈ H and λ ∈ R. The norm of A is defined as

‖A‖ = sup{‖Ax‖ : x ∈ H, ‖x‖ = 1}.

The set of bounded real linear operators is denoted by B(H). Complex linearand antilinear operators represent two extreme cases of real linear operators.An operator A is complex linear, resp. antilinear, if

A(λx) = λAx, resp. A(λx) = λAx,

for all x ∈ H and λ ∈ C. Complex multiplication not being commutative,B(H) is merely a real Banach algebra. Bounded complex linear operators andbounded antilinear operators are real linear subalgebras thereof.

Real linear operator theory and its applications 3

Of course, the theory of complex linear operators is extensive. On theother hand, real linear operators are abundant in applications, too. Classi-cally, antilinear operators occur in quantum mechanics in the study of timereversal [11, p. 250]. More recent examples are described in what follows.

Example 1. The Beltrami equation (1.1) with ν = 0 is central to the twodimensional Calderon problem in the inverse problem of finding L∞-conduc-tivities of a material. There, in the construction of complex geometrical opticssolutions, the real linear operator

A = P (I − βτS)−1ατ (2.1)

is translated as λI−A which then needs to be inverted. Here P is the Cauchytransform, S the Beurling transform, α ∈ L∞(C) with support in the unitdisk D, and β ∈ L∞(C) with |β(z)| ≤ kχD(z) for almost all z ∈ C with aconstant 0 < k < 1. For more details, see [3, Section 4].

Example 2. Purely antilinear operators arise in the study of inverse scatteringand nonlinear evolution equations. Namely, the so-called ∂-equation

∂v(k) = −T (k)τv(k) (2.2)

where v : R2 → C is assumed to satisfy the condition lim|k|→∞ v(k) = 1and T : R2 → C to be compactly supported, is then studied. Eq. (2.2) isequivalent to

(I + C 1πkMT τ)v = 1, (2.3)

where 1 is the function having constant value 1, C 1πk

denotes convolution by1πk and MT the multiplication by T . For further applications and numericalapproximation, see [16] and references therein.

2.2. The complex linear–antilinear representation

Every real linear operator A can be represented uniquely as the sum of acomplex linear and an antilinear operator. Namely, we have

A = A0 +A1 (2.4)

with A0 = 12 (A− iAi) complex linear and A1 = 1

2 (A+ iAi) antilinear. Thus,B(H) can be regarded as having the structure of a Z2-graded unital algebraover R, i.e. a unital superalgebra over R, where the even elements are complexlinear and odd elements are antilinear operators [7, p. 46].

Definition 2.1. An antilinear element κ ∈ B(H) satisfying κ2 = I is said tobe an abstract conjugation.1

On any separable Hilbert space H there exists an abstract conjugationby setting κ = ι−1τι, where ι : H → l2 is a Hilbert space isomorphism and τis the complex conjugation on l2.

Using the representation (2.4) with an abstract conjugation κ, any reallinear operator A can be decomposed as

A = A0 +B0κ (2.5)

1We do not insist on κ being an isometry which is occasionally required [9].


with A0 and B0 = A1κ complex linear. Although appealing, an abstractconjugation may not be natural since it is not unique and depends on thechosen basis of H. Thus, the complex linear–antilinear representation (2.4)can be regarded as canonical in general. In self-conjugate Hilbert spaces, thereis a natural decomposition involving the standard complex conjugation.

Definition 2.2. A Banach space of complex valued functions is called self-conjugate if it is closed under the complex conjugation τ .2

In self-conjugate spaces, aside from (2.4), a canonical decomposition ofa real linear operator A is

A = A0 +B0τ (2.6)

with unique complex linear operators A0 and B0.The representation (2.6) is not always possible. Even in realistic applica-

tions one must accept dealing with the decomposition (2.4). This is in strongcontrast with real linear matrix analysis where the representation (2.6) isalways available [13].

Example 3. The so-called Friedrichs operator F of a planar domain D isdefined on A2(D), the Bergman space of square-integrable analytic functionson D [20], [8]. Two basic problems in planar elasticity correspond to findingelements u, v ∈ A2(D) such that

(I + F )u = f, resp. (kI − F )v = g, (2.7)

for some given f, g ∈ A2(D) and a material constant k ∈ R. The Friedrichsoperator on A2(D) is purely antilinear being defined as F = Pτ , where Pis the orthogonal projection P : L2(D) → A2(D). Because the Bergmanspace A2(D) is not self-conjugate, the representation (2.6) does not exist forthe Friedrichs operator. It is noteworthy that whether F is of finite rank orcompact, depends on D. For instance, it is known that when D is simplyconnected and has a C1,α boundary, F is compact [17].

In view of the preceding example, a Hilbert space H consisting of com-plex valued functions can be considered as a subspace of the closure of H+τHdenoted here by X. Let us define F = Pτ : H → H, where P is the orthog-onal projection on X onto H. The antilinear operator F can be regarded tomeasure how self-conjugate H is by the fact that H is self-conjugate if andonly if the range of F is H. For example, for A2(D), where D is the unit disc,F is of rank 1, and for so-called quadrature domains, it is of finite rank [20].These Hilbert spaces can be regarded as being very far from self-conjugate.

2Any Banach space is isometrically isomorphic to a norm closed subspace of C(X), the

space of complex valued continuous functions on a compact Hausdorff space, endowed withthe supremum norm.


2.3. The adjoint of a real linear operator

Denote by (·, ·) the inner product in H. The adjoint A∗0 of a complex linearoperator A0 is defined as usual by the condition

(A0x, y) = (x,A∗0y) for all x, y ∈ H.Define the adjoint A∗1 of an antilinear operator A1 by

(A1x, y) = (x,A∗1y) for all x, y ∈ H.Then, for a real linear operator A split according to (2.4), we define its adjointby

A∗ = A∗0 +A∗1.

This makes B(H) into a real Banach ∗ algebra, i.e. a real Banach algebrawith a real linear involution ∗ satisfying

A∗∗ = (A∗)∗ = A and (AB)∗ = B∗A∗

for all A,B ∈ B(H).3

Definition 2.3. A real linear operator A is called self-adjoint if A = A∗.

For example, the Friedrichs operator in (2.7) is self-adjoint.

2.4. Spectral theory

Although we are dealing with a real linear Banach algebra, complex translatesof real linear operators appear regularly in applications. (See the examplesabove.) In view of this, we set the following definition.

Definition 2.4. The spectrum σ(A) ofA ∈ B(H) is the set of complex numbersλ, called spectral values, for which λI −A is not boundedly invertible.

We call λ ∈ C an eigenvalue of A if there exists a nonzero x ∈ H suchthat Ax = λx. The point spectrum σp(A) of A consists of the eigenvalues ofA.

Definition 2.5. [21] The lower bound of an operator A ∈ B(H) is

m(A) = inf{‖Af‖ : ‖f‖ = 1, f ∈ H}.

A complex number λ is said to be in the approximate point spectrumσa(A) of A if m(λI − A) = 0. A complex number λ is in the compressionspectrum σc(A) of A if the closure of ran(λI −A) differs from H, i.e. λ is aneigenvalue of A∗. We have σ(A) = σa(A) ∪ σc(A).

It is noteworthy that the spectrum can be empty since B(H) is merely

a real Banach algebra. A finite dimensional example is given by[

0 1−1 0

]τ ∈

B(C2). It is not straightforward to give conditions guaranteeing the existenceof the spectrum.

It follows from the following proposition that the spectrum of a boundedreal linear operator A is compact.

3The adjoint can be shown to satisfy Re(Ax, y) = Re(x,A∗y) for all x, y ∈ H.


Proposition 2.6. Assume A = B + C ∈ B(H) with B invertible and ‖C‖ <m(B). Then A is invertible.

Proof. We can factor A = B(I+B−1C). By using the Neumann series we canconclude that the operator I + B−1C is invertible since ‖B−1C‖ ≤ ‖C‖

m(B) <

1. �

Proposition 2.7. For any µ ∈ C \ σ(A) there holds1

dist(µ, σ(A))≤ ‖(µI −A)−1‖.

Proof. Let α ∈ C be such that |α| < ‖(µI − A)−1‖−1. Then by using theNeumann series, (α+µ)I −A = (µI −A)[(µI −A)−1α− I] is invertible andα+ µ ∈ C \ σ(A) from which the result follows. �

In the complement of σ(A), the resolvent is defined as R(λ;A) = (λI −A)−1. For |λ| > ‖A‖, it can be expressed in terms of the Neumann series as

R(λ;A) = λ−1∞∑k=0

(λ−1A)k.

(Hence, maxλ∈σ(A) |λ| ≤ ‖A‖.) The spectrum of the resolvent can be foundby using the following proposition.

Proposition 2.8. Suppose A ∈ B(H) be a real linear operator and f(z) =(az + b)(cz + d)−1, with a, b, c, d ∈ C. If −d/c is not a spectral value of A,then

f(σ(A)) = σ(f(A)).

Proof. For a polynomial p of degree at most one, we have p(σ(A)) = σ(p(A)).Moreover, from λI −A = λ(A−1 − λ−1I)A, it follows that if A is invertible,then σ(A−1) = 1/σ(A). Combining these observations with

λI − (aA+ bI)(cA+ dI)−1 = a

[c

aλI − I −

(b

a− c

d

)(d

cI +A

)−1]

1c

yields the result. �

For functions that are not linear fractional transformations, the spectralmapping theorem fails in general.

For the adjoint operator there holds σ(A) = σ(A∗) by the facts that theoperation of taking the adjoint ∗ is additive and that a real linear operatoris invertible if and only if its adjoint is.

The spectrum is preserved in the orbit of complex linear operators.

Proposition 2.9. For an operator A ∈ B(H), set

O(A) = {SAS−1 : S complex linear and invertible in B(H)}.Then the spectrum of each element in O(A) coincides with that of A.

Proposition 2.10. The spectrum function is upper semi-continuous.


This is proved similarly to the complex linear case; see e.g. [4, p. 50].Since the spectrum may be empty, let us precise: if An → A in operator normand σ(A) = ∅, then σ(An) = ∅ for sufficiently large n.

Theorem 2.11. An operator A ∈ B(H) has a real spectral value r if and onlyif r is in the spectrum of AC, a canonical complexification of A. Furthermore,

σp(A)∩R = σp(AC)∩R, σa(A)∩R = σa(AC)∩R, σc(A)∩R = σc(AC)∩R.Proof. With respect to an abstract conjugation κ, define the complexificationof A as4

AC : H ⊕H → H ⊕H :(x1

x2

)7→(A BB A

)(x1

x2

), (2.8)

where A = A+Bκ is split according to the representation (2.5). Let us denotefor brevity A = κAκ and x = κx. Clearly, AC is a bounded complex linearoperator on H ⊕H. Note that we have (A∗)C = (AC)∗.

For a vector of the form (x, x) ∈ H ⊕H, we have AC(x, x) = (Ax,Ax).Let Ax = rx with x 6= 0. Then AC(x, x) = (Ax,Ax) = r(x, x), so thatσp(A) ∩R ⊂ σp(AC) ∩R. Reasoning similarly, we get the inclusions σa(A) ∩R ⊂ σa(AC) ∩ R and σc(A) ∩ R ⊂ σc(AC) ∩ R.

Suppose then that r ∈ σp(AC)∩R, i.e. there is a nonzero (x, y) ∈ H⊕Hsuch that (

A BB A

)(xy

)=(Ax+ByBx+Ay

)= r

(xy

).

Adding the conjugated bottom row to the top row, we get A(x+ y) +B(x+y) = r(x + y), i.e. A(x + y) = r(x + y). If x = −y, we have from the toprow that Ax − Bx = rx, whence A(ix) = A(ix) + Bκ(ix) = r(ix). Thusr ∈ σp(A) ∩ R.

Assume r ∈ σa(AC)∩R, i.e. m(rI −AC) = 0. This is equivalent to thatthere is a sequence of vectors (xn, yn) ∈ H ⊕ H such that ‖(xn, yn)‖⊕ ≥ εfor some ε > 0 and∥∥∥∥(A− rI B

B A− rI

)(xnyn

)∥∥∥∥⊕

=∥∥∥∥((A− rI)xn +BynBxn + (A− rI)yn

)∥∥∥∥⊕→ 0.

Since ‖(x, y)‖2⊕ = ‖x‖2 + ‖y‖2, both ‖(A− rI)xn +Byn‖ and ‖(A− rI)yn +Bxn‖ = ‖Bxn + (A− rI)yn‖ tend to zero. Hence,

‖(A−rI)(xn−yn)+B(xn−yn)‖ ≤ ‖(A−rI)xn+Byn‖+‖(A−r)yn+Bxn‖ → 0.

If the sequence (xn + yn) is bounded from below by some positive constant,the claim follows. If not, then ‖xn + yn‖ → 0. Since ‖(xn, yn)‖⊕ is boundedfrom below, either (xn) or (yn) contain a subsequence that is bounded frombelow by some positive constant. With no loss of generality, assume that(xnk) is that subsequence. Then

‖(A− rI)xnk −Bxnk‖ ≤ ‖(A− rI)xnk +Bynk‖+ ‖ −Bynk −Bxnk‖ → 0,

so that ‖(A− rI)ixnk +Bixnk‖ → 0 showing that r ∈ σa(A) ∩ R.

4The direct sum Hilbert space is equipped as usual with the inner product (x, y)⊕ =(x1, y1)H + (x2, y2)H .


Assume r ∈ σc(AC) ∩ R, i.e. r is an eigenvalue of (AC)∗ Since

(A∗)C = (AC)∗ =(A∗ B

∗

B∗ A∗

),

reasoning similarly as above with the eigenvalues we can conclude that r ∈σc(AC) ∩ R. �

Although the spectrum of a self-adjoint real linear operator need not lieon the real axis, we do have the following corollary.

Corollary 2.12. If A is self-adjoint, then σ(A)∩R is nonempty and the spec-trum is symmetric with respect to the real axis.

Proof. If A is self-adjoint, then also the complex linear AC is self-adjoint.Thus σ(AC) is real and nonempty and by Theorem 2.11 σ(A)∩R is nonempty.Since σ(A) = σ(A∗), the spectrum of A is symmetric with respect to the realaxis. �

This combined with Proposition 2.8 yields the following corollary.

Corollary 2.13. Assume A = (aB+bI)(cB+dI)−1 for a self-adjoint B ∈ B(H)with a, b, c, d ∈ C. Then the spectrum of A is nonempty.

Purely antilinear operators appear often in applications. (See Examples2 and 3 above.) For them we have the following propositions.

Proposition 2.14. Let A be antilinear. Then the point spectrum σp(A), theapproximate point spectrum σa(A) and the compression spectrum σc(A) arecircularly symmetric with respect to the origin.

Proof. Let λ ∈ σa(A), i.e. there is a sequence of unit vectors (fn) such that‖Afn − λfn‖ → 0. Then for any θ ∈ R, e2θiλ is an approximate eigenvaluewith approximate eigenvectors e−θifn which can be seen from

‖A(e−θifn)− e2θi(e−θifn)‖ = ‖eθi(Afn − λfn)‖ = ‖Afn − λfn‖.From this it can be seen that the point spectrum is circularly symmetric.Hence the point spectrum of A∗ is also circularly symmetric which impliesthat the compression spectrum is circularly symmetric. �

Observe that if A is antilinear, then A2 is complex linear.

Proposition 2.15. Let A be antilinear. Then λ ∈ σ(A) if and only if |λ|2 ∈σ(A2).

Proof. By Proposition 2.14, we can assume that λ = |λ|.Let λI − A be boundedly invertible where λ ∈ R. Then by symmetry

λI +A is also boundedly invertible, and thus (λI −A)(λI +A) = λ2I −A2

is boundedly invertible.On the other hand, if λ ∈ σa(A), then there is a sequence of unit

vectors (fn) for which ‖(λI − A)fn‖ → 0. Then also ‖(λ2I − A2)fn‖ ≤‖λI+A‖‖(λI−A)fn‖ → 0. If λ ∈ σc(A), then there is a nonzero vector f forwhich (λI −A∗)f = 0, and thus (λ2I − (A∗)2)f = 0. Hence λ2 ∈ σc(A). �


Proposition 2.16. If A is antilinear and satisfies A = −A∗, then σ(A) ⊂ {0}.

Proof. By Proposition 2.14, we need to consider only non-negative real num-bers. Since A2 = −A∗A is complex linear self-adjoint and for all x ∈ H wehave (A∗Ax, x) = (Ax,Ax) = (Ax,Ax) ≥ 0, the spectrum of A2 lies in thenon-positive real axis. Thus A2−λ2I = (A−λI)(A+λI) is invertible for allλ > 0. Thereby only the origin can be in the spectrum of A. �

For complex linear operators A and B on H, it holds that

σ(AB) \ {0} = σ(BA) \ {0}

This stems from the ring-theoretic assertion that, if I−AB is invertible, thenI −BA is invertible [10]. For real linear operators an analogy is as follows.

Proposition 2.17. Let A be real linear and B complex linear bounded operators.Then

σ(AB) \ {0} = σ(BA) \ {0}.

Proof. Let λ 6= 0. Then AB0 − λI = (ABλ−1 − I)λ = (Aλ−1B − I)λ isinvertible if and only if (BAλ−1−I)λ is invertible, i.e. (BA−λI) is invertible.

�

These conditions cannot be relaxed in general. Only if B is taken fromthe other extreme, we have the following identity.

Proposition 2.18. Let A be real linear and B antilinear bounded operators.Then

σ(AB) \ {0} = σ(BA) \ {0}.

Proof. Let λ 6= 0. ThenAB−λI = (ABλ−1−I)λ = (Aλ−1B−I)λ is invertibleif and only if (BAλ−1 − I)λ is invertible, i.e. (BA− λI) is invertible. �

Corollary 2.19. Let A ∈ B(H) be antilinear. Then σ(A2) is symmetric withrespect to the real axis.5

Proof. Set A = B in Theorem 2.18. �

Example 4. TakeA = ντS, where S is the Beurling transform. The norm ofAis typically used to establish invertibility of I−ντS based on the convergenceof the Neumann series. For sharper estimates, in view of Proposition 2.15, it isadvisable to study the spectrum of the complex linear operator A2 = νS∗νS.

5Propositions 2.14, 2.15, 2.6, 2.17, 2.18 and Corollary 2.19 can be shown to hold as suchfor bounded real linear operators on a complex Banach space.


2.5. Compact and unitary operators

In complex linear operator theory, compact operators are regarded as ’small’and unitary operators as ’large’. Thus, constituting two extremes, compactand unitary operators provide extensively studied classes of operators.

For compactness the real linear analogue is straightforward.

Definition 2.20. A real linear operator A = A0 + A1 split according to (2.4)is said to be compact if A0 and A1 are compact.6

The spectrum of a compact operator can contain a continuum [13].However, it cannot have interior points by being small in the following sense.

Theorem 2.21. Assume A ∈ B(H) is compact and dimH =∞. Then(i) for any line L passing through the origin, σ(A)∩L is countable contain-

ing the origin with the origin being its only possible limit point, and(ii) every spectral value λ 6= 0 is an eigenvalue.

Proof. We can consider only the real line R, since if the conclusion holds forR, it holds also for any line {reiθ : r ∈ R, with θ ∈ [0, 2π) fixed} by factoringA− reiθI = eiθ(e−iθA− rI).

If A = A+ Bκ is compact, then so is the complexification AC given inEq. (2.8) by the following arguments. Let (hn)n∈N ⊂ H ⊕ H be a boundedsequence, where hn = (xn, yn). Then also (xn) and (yn) are bounded se-quences. Since A is compact, taking a subsequence four times, we may finda subsequence (hn)n∈I , I ⊂ N, such that (Axn), (Byn), (Bxn) and (Ayn)converge. Thus also (AChn)n∈I converges, and AC is compact.

SinceAC is a compact complex linear operator, its spectrum is countablewith the origin being its only possible limit point and every nonzero spectralvalue is an eigenvalue. The result follows from the fact σ(A) ∩ R = σ(AC) ∩R. �

In the infinite dimensional case, the spectrum of A can vanish only ifA1 is large.

Corollary 2.22. Assume dimH =∞. For A ∈ B(H) with A1 compact, σ(A)is nonempty.

Proof. The result follows from the facts that the essential spectrum of acomplex linear operator is always non-empty and compact, and that theessential spectrum is invariant under compact perturbations [18]. �

Example 5. This is Example 1 continued. It is known that A defined in (2.1)is compact on Lp(C) with 2 < p < 1

k and that its spectrum lies in the closeddisk centered at origin with radius k (cf. [3] and Proposition 4.1 in particular).That the spectrum is actually nonempty follows from Corollary 2.22.

6This is the same as requiring the image under A of any bounded set in H be relativelycompact. Equivalently, for any bounded sequence (fn) the sequence (Afn) contains a

convergent subsequence. Hence, we have a natural extension of the complex linear notionof compactness.


In complex linear operator theory, the way to assess compactness andapproximate the spectrum of a compact operator is based on using finite rankapproximants. Analogously, we set the following.

Definition 2.23. An operator A ∈ B(H) is said to be of finite rank, if it canbe given as the finite sum

A =n1∑k=1

(·, uk)vk +n2∑k=1

(uk, ·)vk (2.9)

for some uk, vk, uj , vj ∈ H, k = 1, . . . , n1, j = 1, . . . , n2.

The spectrum of finite rank operators is well understood.

Theorem 2.24. Assume dimH = ∞. For an operator A ∈ B(H) given asin (2.9), σ(A) is the union of the origin and a real algebraic plane curve ofdegree 2(n1 + n2) at most.

Proof. Since the range of A is contained in a finite dimensional and henceproper subspace of H, the origin must be in σ(A).

The operator A maps H1 = span{uj , uk : j = 1, . . . , n1, k = 1, . . . , n2}into H2 = span{vj , vk : j = 1, . . . , n1, k = 1, . . . , n2}, both subspaces ofdimension n1 + n2 at most. Then A = A0 ⊕ 0 with A0 : span{H1, H2} →span{H1, H2}. Hence, the spectral problem is finite dimensional. We haveA0x = λx if and only if x ∈ H2. Thereby we can restrict A0 to H2. Supposethis restriction is represented as A0 + B0τ with A0, B0 ∈ C(n1+n2)×(n1+n2).For a nonzero x ∈ C(n1+n2) it holds A0x− λx = 0 if and only if[

A0 − λI B0

B0 B0 − λI

] [xx

]= 0 ⇐⇒ det

[A0 − λI B0

B0 B0 − λI

]= 0.

This determinant is a polynomial in λ and λ and thus its zero set is a realalgebraic curve of degree 2(n1 + n2) at most. �

With finite rank approximations to a compact real linear operator A,Proposition 2.10 and Corollary 2.7 can be used to estimate the spectrum.Although not considered in this paper, there are at least three ways to con-struct approximations. Certainly, with finite rank approximations to A0 andA1, their sum yields a real linear finite rank approximation to A. However,this is a crude approximation that merely separates the approximation prob-lem in a simple manner. It appears more natural and effective to study ap-proximations in terms of right and left approximation numbers.

Definition 2.25. [12] For A ∈ B(H), define its right approximation numbersas

σn(A) = inf{‖A(I − P )‖ : P = P ∗ is complex linear, P 2 = I, rankP ≤ n}(2.10)

and its left approximation numbers as σn(A∗).


For a classical extension of complex linear unitary operators, a reallinear operator A is called a symmetry of H if

|(Ax,Ay)| = |(x, y)|

for every x, y ∈ H [22]. Wigner’s theorem states that every symmetry iseither complex linear and unitary or an antilinear bijective isometry [5]. Thischaracterizes symmetries of a Hilbert space thoroughly. This extension is toorestrictive for our purposes. For an appropriate generalization, we set thefollowing definition.

Definition 2.26. A real linear operator A is called unitary if A is a bijectiveisometry.

We denote the group of unitary real linear operators by U(H).

Example 6. Assume A = −A∗. (Then λ ∈ σ(A) ⇐⇒ −λ ∈ σ(A). ) Itsexponential defined as eA =

∑∞k=0

Akk! is unitary.

Proposition 2.27. The spectrum of a unitary operator A ∈ B(H) is a subsetof the unit circle.

Proof. The origin cannot be in the spectrum, since A is invertible. For aunitary A, we have ‖A‖ = m(A) = 1. If |λ| < 1, then by Proposition 2.6λI −A is invertible. If |λ| > 1, then it follows that λI −A = λ(I − λ−1A) isinvertible by using the Neumann series. �

It is noteworthy that the spectrum of a unitary real linear operator canbe empty.

3. Real linear multiplication operators on self-conjugateHilbert spaces

In this section, motivated by the structure of the Beltrami operator, we studya real linear analogue of multiplication (Laurent) operators and their unitaryapproximation.

3.1. Basic properties

Recall that the Beurling transform is given as a singular integral operator ofCalderon–Zygmund type

Sf(z) = − 1π

limε→0

∫|z−w|>ε

f(w)(z − w)2

dw1dw2, w = w1 + iw2,

for f ∈ L2(C). The Beurling transform is unitary which can be seen from theuseful identity [1, p. 105]

F [Sf ](ξ) =ξ

ξF [f ](ξ), (3.1)


where F denotes the planar Fourier transform

F [f ](z) =∫

Ce−iπ(zw+zw)f(w) dw1dw2.

From Eq. (3.1) it follows that the spectrum of S is the whole unit circle.Consider the Beltrami operator

I − (µ+ ντ)S. (3.2)

Since the Beurling transform is a Fourier multiplier by Eq. (3.1), we have

F(µ+ ντ)SF∗ = F(µ+ ντ)F∗m, (3.3)

where m(z) = z/z and m denotes also the multiplication operator mf(ξ) =ξξf(ξ). Here F , F∗ and m are complex linear and unitary. This is an intriguingstructure as the operator given by Eq. (3.3) is complex linearly unitarilyequivalent to µ+ ντ .7

In what follows, assume D ⊂ C is a domain, µ, ν ∈ L∞(D) and 1 ≤ p ≤∞ unless otherwise stated. Denote by χA the characteristic function of a setA.

Definition 3.1. A real linear multiplication operator µ+ ντ is a mapping onLp(D) defined by

f 7→ (µ+ ντ)f = µf + νf.

Operators of this type can be viewed as being a real linear generalizationof the classical multiplication operators, also called Laurent operators, oper-ating on self-conjugate Hilbert spaces. Being the starting point for Toeplitzoperator theory, multiplication operators are of central relevance. (See thehighly cited paper [6].)

Real linear multiplication operators give also rise to multipliers.

Definition 3.2. An operator A on L2(C) is a real linear Fourier multiplier ifit can be given as A = F∗(µ+ ντ)F .

Clearly, a real linear multiplication operator µ + ντ is real linear andbounded on Lp(D). Its norm is given as follows.

Theorem 3.3. The norm of µ+ τν on Lp(D) is ‖|µ|+ |ν|‖∞.

Proof. Let p <∞ and f ∈ Lp(D). We have∫D

|µ(z)f(z) + ν(z)f(z)|p dA(z) ≤∫D

(|µ(z)|+ |ν(z)|)p|f(z)|p dA(z)

≤∫D

‖|µ|+ |ν|‖p∞|f(z)|p dA(z)

= ‖|µ|+ |ν|‖p∞‖f‖ppand therefore ‖µ+ ντ‖ ≤ ‖|µ|+ |ν|‖∞.

7Operators A,B ∈ B(H) are complex linearly unitarily equivalent if there exist complexlinear unitary operators U0 and V0 such that A = U0BV0.


Set C = ‖|µ|+ |ν|‖∞, Uε = (|µ|+ |ν|)−1(B(C, ε)) (the preimage of theball B(C, ε) under the mapping |µ| + |ν|), and mε = A(Uε) the measure ofUε. Since C is in the essential range of |µ|+ |ν|, mε is positive for all ε > 0.Let us define

fε(z) = ei(arg ν(z)−arg µ(z))

2 )χUε(z)m−1ε .

Then we have that

‖fε‖pp =∫D

|ei(arg ν(z)−arg µ(z))

2 )χUε(z)m−1ε |p dA(z) = 1

and that

|µ(z)fε(z) + ν(z)fε(z)| = ||µ(z)|ei arg µ(z)ei(arg ν(z)−arg µ(z))

2 )

+ |ν(z)|ei arg ν(z)e−i(arg ν(z)−arg µ(z))

2 )|χUε(z)m−1ε

= ||µ(z)|e i2 (arg µ(z)+arg ν(z))

+ |ν(z)|e i2 (arg µ(z)+arg ν(z))|χUε(z)m−1ε

= (|µ(z)|+ |ν(z)|)χUε(z)m−1ε

≥ (C − ε)χUε(z)m−1ε .

Thus ∫D

|µ(z)fε(z) + ν(z)fε(z)|p dA(z) ≥∫

C(C − ε)pχUε(z)m−1

ε dA(z)

= (C − ε)p −→ε→0

Cp

and therefore ‖µ+ ντ‖ = ‖|µ|+ |ν|‖∞.For p =∞, we reason similarly. We have

‖µf + νf‖∞ = ess supD|µf + νf | ≤ ess sup

D(|µ|+ |ν|)|f | ≤ ‖|µ|+ |ν|‖∞‖f‖∞.

Let us define fε(z) = ei(arg ν(z)−arg µ(z))

2 )χUε(z). With a calculation similar tothe one above, we get

ess supD|µfε + νfε| ≥ ess sup

Uε

(C − ε) = C − ε −−−→ε→0

C.

Therefore ‖µ+ ντ‖ = ‖|µ|+ |ν|‖∞ on L∞(D). �

A real linear multiplication operator µ+ ντ is algebraically invertible if|µ(z)| 6= |ν(z)| a.e. on D. Its algebraic inverse is the real linear multiplicationoperator

1|µ|2 − |ν|2

(µ− ντ).

This follows by direct calculation

(µ− ντ)(µ+ ντ) = |µ|2 − ντµ+ µντ − ντντ= |µ|2 − νµτ + µντ − νντ2

= |µ|2 − |ν|2.

Whenever |µ(z)| 6= |ν(z)| a.e., we can divide this equation by |µ|2 − |ν|2.


Proposition 3.4. If zero is not in the essential range of |µ|2 − |ν|2, then theinverse (µ+ ντ)−1 is bounded on Lp(D).

Proof. Assume zero is not in the essential range of |µ|2 − |ν|2. Then thereexists ε > 0 such that the measure of the preimage of the ball B(0, ε) is zero.Hence, multiplication by (|µ|2 − |ν|2)−1 is bounded by 1/ε and the resultfollows. �

The condition on invertibility can be cast in a different form by using thelower bound of an operator. For the lower bound of a real linear multiplicationoperator we have the following.

Theorem 3.5. The lower bound of a real linear multiplication operator µ+ντis

m(µ+ ντ) = ess infD||µ| − |ν||.

Proof. Just as in the proof of Theorem 3.3, set C = ess inf ||µ| − |ν||, Vε =(||µ| − |ν||)−1(B(C, ε)) (the preimage of the ball B(C, ε) under the mapping||µ| − |ν||), and mε = A(Vε) the measure of Vε. Since C is in the essentialrange of ||µ| − |ν||, mε is positive for all ε > 0. Let us define

fε(z) = ei(arg ν(z)−arg µ(z))+π

2 )χVε(z)m−1ε .

We have

‖fε‖pp =∫D

|ei(arg ν(z)−arg µ(z)+π)

2 )χVε(z)m−1ε |p dA(z) = 1

and

|µ(z)fε(z) + ν(z)fε(z)| = ||µ(z)|ei arg µ(z)ei(arg ν(z)−arg µ(z)+π)

2 )

+ |ν(z)|ei arg ν(z)e−i(arg ν(z)−arg µ(z)+π)

2 )|χVε(z)m−1ε

= ||µ(z)|e i2 (arg µ(z)+arg ν(z)+π)

+ |ν(z)|e i2 (arg µ(z)+arg ν(z)−π)|χVε(z)m−1ε

= (||µ(z)| − |ν(z)||)χVε(z)m−1ε

≤ (C + ε)χVε(z)m−1ε .

Hence∫D

|µ(z)fε(z) + ν(z)fε(z)|p dA(z) ≤∫D

(C + ε)pχUε(z)m−1ε dA(z)

= (C + ε)p −→ε→0

Cp.


On the other hand, let f ∈ Lp(D) with ‖f‖p = 1. By the triangle inequality,

‖µf + νf‖ ≥ |‖µf‖ − ‖νf‖

=

{‖µf‖ − ‖νf‖−‖µf‖+ ‖νf‖

≥

{ess inf |µ| − ess sup |ν|− ess sup |µ|+ ess inf |ν|

=

{ess inf |µ|+ ess inf −|ν|ess inf −|µ|+ ess inf |ν|

= ess inf ||µ| − |ν||�

As a corollary to Proposition 3.4 and to Theorem 3.5, we get a charac-terization for the resolvent set of a real linear multiplication operator.

Corollary 3.6. A complex number λ is not in the spectrum of a real linearmultiplication operator µ + ντ if and only if there is a constant C > 0 suchthat ||µ− λ| − |ν|| > C a.e. on D.

Proof. Assume 0 is not in the essential range of

|µ|2 − |ν|2 = (|µ|+ |ν|)(|µ| − |ν|).Then it cannot be in the essential range of |µ| + |ν| or |µ| − |ν|. The latterimplies that there is a constant C > 0 such that ||µ| − |ν|| > C a.e. onD. The greatest constant is of course ess infD ||µ| − |ν||. Conversely, if 0 <ess inf ||µ| − |ν||, then zero is not in the essential range of |µ| − |ν| nor of|µ|+ |ν|. �

Finally, first observe that a real linear multiplication operator µ + ντis self-adjoint if and only if µ is real valued. Second, there are no non-trivialcompact multiplication operators. The structure of unitary real linear multi-plication operators is the subject of the next paragraph.

3.2. Unitary approximation

The unitary approximation problem of a complex linear operator is typicallyconnected to energy conservation and probability. (For the unitary approxi-mation, see [21].) Another related problem is that of approximating a complexlinear operator with a scalar multiple of a unitary operator. Motivated by theBeltrami equation, we will board these subjects in a special case.

The invertibility of the Beltrami operator

I − (µ+ ντ)S

is determined by the spectrum of (µ + ντ)S. By the fact that the Beurlingtransform S is itself complex linear and unitary, it is natural to view (µ +ντ)S as a perturbation of a real linear unitary operator. Next such a unitaryoperator is constructed.


Proposition 3.7. Unitary real linear multiplication operators are of the form

χA(z)eiφ(z) + χD\A(z)eiψ(z)τ,

where A ⊂ D is a measurable subset, and φ, ψ are some measurable real-valued functions.

Proof. Suppose µ+ ντ is unitary and there is a ball B such that |µ| > 0 and|ν| > 0 a.e. on B. We have

(µf + νf)(µf + νf) = (|µ|2 + |ν|2)|f |2 + 2 Reµνf2.

Set f(z) = eiπ/4a(z)e−i arg µ/2ei arg ν/2

(|µ||ν|)1/2 χB , where a is real valued. Then

Reµνf2 = 0

and

‖f‖2 =∫B

a2

|µ||ν|= ‖(µ+ντ)f‖2 =

∫B

(|µ|2+|ν|2)|f |2 =∫B

(1|ν|2

+1|µ|2

)a2

which leads to ∫B

(1|µ|− 1|ν|

)2

a2 = 0.

Since a was arbitrary, we see that |µ| = |ν| a.e. on B, but this contradictsinvertibility of µ+ντ . Hence we must have µ+ντ = χAµ+χD\Aντ for someA ⊂ D.

If 0 < |µ| < 1 a.e. on some B ⊂ D, then taking f = χBg, with anarbitrary g ∈ L2(D), we have

‖(µ+ ντ)f‖2 =∫B

|µ|2|f |2 <∫B

|f |2 = ‖f‖2

which contradicts the isometricity of µ+ ντ . Reasoning similarly in the case|µ| > 1 and for the antilinear part ν, we have the result. �

Observe that if a unitary real linear multiplication operator is complexlinear, i.e. if the set D \ A has measure zero, then its spectrum is given bythe classical complex linear theory as the essential range of eiφ. On the otherhand, if the set D\A has positive measure, then by Corollary 3.6 the spectrumis the whole unit circle irrespective of the essential ranges of φ or ψ.

The following theorem by Rogers is needed in what follows.

Theorem 3.8. [21, 19] Let T be a complex linear operator with indT = 0.8

Then its distance from the set of complex linear unitary operators is

max{‖T‖ − 1, 1−m(T )}

with the lower bound of T being m(T ) = inf(σ(|T |)).

This allows us to approximate (µ+ ντ)S with a unitary operator.

8The index of an operator A on a Hilbert space H is defined as indA = dim kerA −dim cokerA, where cokerA = H/ ranA.


Theorem 3.9. Assume µ + ντ is an invertible real linear multiplication op-erator and B is complex linear unitary. Then the best real linear unitaryapproximant of (µ+ ντ)B is

U = (χAei arg µ + χC\Aei arg ντ)B

giving the distance

‖(µ+ ντ)B − U‖ = max{ess sup |µ|+ |ν| − 1, 1− ess inf ||µ| − |ν||}.

Proof. Let µ, ν ∈ L∞ with µ(z) = |µ(z)|ei arg µ(z) and ν(z) = |ν(z)|ei arg ν(z).Define

A = {z ∈ C : ∃ε > 0 s.t. |1− |µ||+ |ν| < |1− |ν||+ |µ| a.e. on B(z, ε)}.

A candidate for the unitary approximant of (µ+ ντ)B is the operator(χAei arg µ + χC\Ae

i arg ντ)B, since

‖(µ+ ντ)B − (χAei arg µ + χC\Aei arg ντ)B‖

≤ ‖µ− χAei arg µ + (ν − χC\Aei arg ν)τ‖

= ‖|µ− χAei arg µ|+ |ν − χC\Aei arg ν |‖∞

≤ max{

ess supA |µ− χAei arg µ|+ |ν − χC\Aei arg ν |

ess supC\A |µ− χAei arg µ|+ |ν − χC\Aei arg ν |

}= max{ess sup

A||µ| − 1|+ |ν|, ess sup

C\A||ν| − 1|+ |µ|}.

Observe that (χAei arg µ + χC\Aei arg ντ)B is indeed unitary because B and

χAei arg µ + χC\Ae

i arg ντ are.As µ + ντ is invertible, the index of (µ + ντ)B is zero and by [21] its

distance from the set of complex unitary operators is

d = max{‖(µ+ ντ)B‖ − 1, 1−m((µ+ ντ)B)}= max{ess sup

C|µ|+ |ν| − 1, 1− ess inf

C||µ| − |ν||}

= max{ess supC

(|µ|+ |ν| − 1), ess supC

(1− ||µ| − |ν||)}.

Then we have

ess supA||µ| − 1|+ |ν| =

{ess supA |µ| − 1 + |ν|ess supA−|µ|+ 1 + |ν|

≤ d

and

ess supC\A

||ν| − 1|+ |µ| =

{ess supC\A |ν| − 1 + |µ|ess supC\A−|ν|+ 1 + |µ|

≤ d.

Thus U = (χAei arg µ+χC\Aei arg ντ)B is the unitary approximant in the case

of an invertible real linear multiplication operator. �

Finding the spectrum of (µ + ντ)S is challenging in general. However,the problem of finding the spectrum of its unitary approximant might beeasier to approach. It is certainly contained in the unit circle. By the fact


that the spectrum is upper semi-continuous, this allows us to make someconclusions about the location of the spectrum of (µ+ ντ)S.

To end this section, let us note that a mere unitary approximation of(µ + ντ)S is not satisfactory since it does not allow for scaling. It is morenatural to approximate (µ+ ντ)S with scaled unitary operators.

Proposition 3.10. Let µ+ντ be a real linear multiplication operator, µ0 +ν0τa unitary real linear multiplication operator and r a non-negative function.Then

‖(µ+ ντ)− r(µ0 + ν0τ)‖ = max{ess supA|ν|, ess sup

C\A|µ|},

where A = {z ∈ C : ∃ε > 0 s.t. |µ| > |ν| a.e. on B(z, ε)}.

Proof. Let A ⊂ C be any measurable subset. Then we have

M = ‖µ+ ντ − r(χAei arg µ + χC\Aei arg ντ)‖

= ‖(|µ| − rχA)ei arg µ + (|ν| − rχC\A)ei arg ντ‖= ess sup

C||µ| − rχA|+ ||ν| − rχC\A|

= max{ess supA||µ| − r|+ |ν|, ess sup

C\A|µ|+ ||ν| − r|}. (3.4)

Choosing A = {z ∈ C : ∃r > 0 s.t. |µ| > |ν| a.e. on B(z, r)} and r(z) =χA|µ|+ χC\A|ν|, we have

M = max{ess supA|ν|, ess sup

C\A|µ|}. �

From (3.4) in the proof of the previous proposition, it is evident howand why the function r and the set A should be chosen to minimize M . Whenr is constrained to being a constant, choosing it and a suitable set A is moresubtle. A crude trial for this is given as follows. Define A = {z ∈ C : ∃r >0 s.t. |µ| > |ν| a.e. on B(z, r)}. Then set

a = min{ess infA|µ|, ess inf

C\A|ν|} and b = max{ess sup

A|µ|, ess sup

C\A|ν|}.

Now define r to be the midpoint and L to be the length of the interval (a, b).With these, we have by invoking Eq. (3.4)

M = max{ess supA||µ| − r|+ |ν|, ess sup

C\A|µ|+ ||ν| − r|}

≤ max{ess supA||µ| − r|+ ess sup

A|ν|, ess sup

C\A|µ|+ ess sup

C\A||ν| − r|}

≤ max{ess supA||µ| − r|, ess sup

C\A||ν| − r|}+ max{ess sup

A|ν|, ess sup

C\A|µ|}

≤ max{ess supC\A

|µ|, ess supA|ν|}+

L

2.


References

[1] Kari Astala, Tadeusz Iwaniec, and Gaven Martin. Elliptic partial differen-tial equations and quasiconformal mappings in the plane. Princeton UniversityPress, Princeton, 2009.

[2] Kari Astala, Jennifer L. Mueller, Lassi Paivarinta, and Samuli Siltanen. Numer-ical computation of complex geometrical optics solutions to the conductivityequation. Applied and Computational Harmonic Analysis, 29:265–299, 2010.

[3] Kari Astala and Lassi Paivarinta. Calderon’s inverse conductivity problem inthe plane. Ann. of Math. (2), 163(1):265–299, 2006.

[4] Bernard Aupetit. A Primer on Spectral Theory. Springer-Verlag, New York,1991.

[5] V. Bargmann. Note on Wigner’s theorem on symmetry operations. Journal ofMathematical Physics, 5(7):862–868, 1964.

[6] Arlen Brown and Paul R. Halmos. Algebraic properties of Toeplitz operators.Journal fur die Reine und Angewandte Mathematik, 213:89–102, 1963/1964.

[7] Pierre Deligne et al. Quantum Fields and Strings: A Course for Mathemati-cians. American Mathematical Society, Providence, 1999.

[8] Kurt Friedrichs. On certain inequalities and characteristic value problems foranalytic functions and for functions of two variables. Transactions of the Amer-ican Mathematical Society, 41(3):321–364, May 1937.

[9] Stephan R. Garcia and Mihai Putinar. Complex symmetric operators and ap-plications. Trans. Amer. Math. Soc., 358:1285–1315, 2006.

[10] Paul R. Halmos. A Hilbert Space Problem Book. Springer-Verlag, New York–Berlin, 2nd edition edition, 1982.

[11] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge UniversityPress, Cambridge, reprinted 1991 edition, 1985.

[12] Marko Huhtanen and Olavi Nevanlinna. Approximating real linear operators.Studia Mathematica, 179(1):7–25, 2007.

[13] Marko Huhtanen and Olavi Nevanlinna. Real linear matrix analysis. In Per-spectives in operator theory, volume 75 of Banach Center Publ., pages 171–189.Polish Acad. Sci., Warsaw, 2007.

[14] Tadeusz Iwaniec and Gaven Martin. Geometric Function Theory and Non-linear Analysis. Oxford University Press, New York, 2001.

[15] Tadeusz Iwaniec and Gaven Martin. The Beltrami equation. Memoirs of theAmerican Mathematical Society, 191(893), 2008.

[16] K. Knudsen, J. Mueller, and S. Siltanen. Numerical solution method for thedbar-equation in the plane. J. Comput. Phys., 198(2):500–517, 2004.

[17] Peng Lin and Richard Rochberg. On the Friedrichs operator. Proceedings ofthe American Mathematical Society, 123(11):3335–3342, November 1995.

[18] Vladimir Muller. Spectral Theory of Linear Operators, volume 139 of OperatorTheory: Advances and Applications. Birkhuser Verlag, Basel, 2003.

[19] Catherine L. Olsen. Unitary approximation. Journal of Functional Analysis,85:392–419, 1989.

[20] Mihai Putinar and Harold S. Shapiro. The Friedrichs operator of a planardomain. In Complex analysis, operators, and related topics, volume 113 of Oper.Theory Adv. Appl., pages 303–330. Birkhauser, Basel, 2000.


[21] Donald D. Rogers. Approximation by unitary and essentially unitary operators.Acta Sci. Math. (Szeged), 39:141–151, 1977.

[22] E. P. Wigner. Gruppentheorie und ihre Anwendungen auf die Quantenmechanikder Atomspektren. Frederick Vieweg und Sohn, Braunschweig, 1931.

Marko HuhtanenAalto UniversityInstitute of MathematicsP.O. Box 11100FI-00076 AaltoFinlande-mail: [email protected]

Santtu RuotsalainenAalto UniversityInstitute of MathematicsP.O. Box 11100FI-00076 AaltoFinlande-mail: [email protected]

real linear operator theory and its applicationsmath.aalto.fi/~mhuhtane/realbeltrami_final.pdfreal...

Documents