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On the numerical range behavior underthe generalized Aluthge transformDavid E. V. Rose a & Ilya M. Spitkovsky ba Department of Pure Mathemetics and Mathematical Statistics ,Centre for Mathematical Sciences, University of Cambridge ,Wilberforce Road, Cambridge, CB3 0WB, Englandb Department of Mathematics , The College of William and Mary ,Williamsburg, Virginia 23187, USAPublished online: 04 Dec 2010.
To cite this article: David E. V. Rose & Ilya M. Spitkovsky (2008) On the numerical range behaviorunder the generalized Aluthge transform, Linear and Multilinear Algebra, 56:1-2, 163-177, DOI:10.1080/03081080701264671
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Linear and Multilinear Algebra, Vol. 56, Nos. 1–2, Jan.–Mar. 2008, 163–177
On the numerical range behavior under thegeneralized Aluthge transform
DAVID E. V. ROSEy and ILYA M. SPITKOVSKY*z
yDepartment of Pure Mathemetics and Mathematical Statistics,Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WB, EnglandzDepartment of Mathematics, The College of William and Mary,
Williamsburg, Virginia 23187, USA.
Communicated by L. Rodman
(Received 31 October 2006; in final form 29 January 2007)
Let A¼UR be the (right) polar decomposition of an n� n matrix A. The generalized Aluthgetransform is then given by ��ðAÞ ¼ R�UR1��. We investigate the behavior of the numericalrange Wð��ðAÞÞ, and in particular its boundary, as a function of �. We obtain complete resultsin the 2� 2 case, when Wð��ðAÞÞ can be described explicitly, and then discuss which of theestablished properties persist in the general setting.
Keywords: Aluthge transform; Numerical range
AMS Subject Classifications: 15A23; 15A60
1. Introduction
Let A be a square complex matrix with a (right) polar decomposition
A ¼ UR: ð1:1Þ
The generalized Aluthge transform of A is defined for � 2 ð0, 1Þ as
��ðAÞ ¼ R�UR1��; ð1:2Þ
*Corresponding author. Email: [email protected]
Linear and Multilinear AlgebraISSN 0308-1087 print/ISSN 1563-5139 online � 2008 Taylor & Francis
http://www.tandf.co.uk/journalsDOI: 10.1080/03081080701264671
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it is easy to see that ��ðAÞ is determined by A uniquely even though the choice of aunitary factor U in (1.1) is not unique for singular matrices A.
We will use the abbreviated notation �(A) for �1=2ðAÞ. We will also sometimesdenote A by �0(A), which is of course in agreement with (1.2) when A is invertible.Along the same lines, let �1ðAÞ ¼ RU. The latter matrix is not necessarily unique(when A is singular) but all its possible values are unitarily similar to A which isgood enough for our purposes.
The transformation A��ðAÞ was introduced by Aluthge in [1], and has sincebeen studied extensively, see i.e. [9], [3], [4], and [15]. The case of general � 2 ð0, 1Þwas also first considered by Aluthge [2], and then in [13], [14], and [5].
Our next proposition lists some known properties of the generalized Aluthgetransform which follow (almost) directly from its definition.
PROPOSITION 1.1 Given an n� n matrix A, the following hold for its generalized Aluthgetransform:
(i) ��ð�AÞ ¼ ���ðAÞ for all �2C.(ii) �ð��ðAÞÞ ¼ �ðAÞ, where �(X ) stands for the spectrum of X.(iii) ��ðV
�AV Þ ¼ V���ðAÞV for all unitary n� n matrices V.(iv) If A is not invertible, then for any � 2 ð0, 1Þ the kernel of A is a reducing subspace of
��ðAÞ.
A deeper result about the (generalized) Aluthge transform deals with the behavior ofthe numerical range. Recall that the latter is defined as
WðAÞ ¼ fhAx, xi : kxk ¼ 1g, ð1:3Þ
where h�, �i is the usual inner product on Cn.
THEOREM 1.2 For all � 2 ½0, 1�,
Wð��ðAÞÞ � WðAÞ: ð1:4Þ
Theorem 1.2 for the regular (� ¼ 1=2) Aluthge transform was proved in [15], and for thegeneral case in [5].
Our next proposition lists basic facts about the numerical range which we shall use.We refer the reader to [7] for a comprehensive treatment of the numerical range.
PROPOSITION 1.3 Given an n� n matrix A, the following properties hold:
(i) Wð�AÞ ¼ �WðAÞ for all �2C.(ii) W(A) is unitarily invariant: WðAÞ ¼ WðU�AUÞ for any unitary n� n matrix U.(iii) If A is unitarily similar to the direct sum of matrices Aj, then
WðAÞ ¼ convfWðAjÞg:
(iv) For normal matrices A, conv�ðAÞ ¼ WðAÞ.(v) The converse of (iv) is true for n � 4.(vi) ReðAÞ � �I for �2R is equivalent to WðAÞ � fz 2 C : ReðzÞ � �g.
164 D. E. V. Rose and I. M. Spitkovsky
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Here and in what follows, we denote by convX the convex hull of the set X, and usethe standard Lowner partial order on the set of n� n hermitian matrices:
A � B if and only if hAx, xi � hBx, xi for all x 2 Cn:
Property (iii) in the case when there is just one matrix Aj (hence coinciding with A)implies that the numerical range is always convex. On the other hand, (iv) follows from(iii) when all Aj are scalars.
Since �0ðAÞ ¼ UR and �1ðAÞ ¼ RU are unitarily similar, part (ii) of Proposition 1.3implies in particular that Wð�0ðAÞÞ ¼ Wð�1ðAÞÞ:
We begin our investigation by proving in section 2 a generalization of Theorem 1.2for 2� 2 matrices which describes the behavior of the numerical range of ��ðAÞ as afunction of �. In section 3, we show that this behavior does not persist in general formatrices of larger size, but does so for a certain class of 3� 3 matrices. Followingthis, in section 4, we investigate the behavior of the boundary of the numerical rangeof the generalized Aluthge transform as a function of �. Finally, in the appendix weinclude for the convenience of the reader self-contained proofs of two technicallemmas, largely following arguments used in [3].
2. The 2T2 case
We begin with a known fact about the numerical range of 2� 2 matrices, an equivalentstatement of Lemma 1.1-1 from [7]. Throughout this article, we shall denote the trace ofa matrix X by Tr(X ).
THEOREM 2.1 For a 2� 2 matrix A, W(A) is the ellipse with foci �1 and �2 at theeigenvalues of A and minor axis of length
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTrðA�AÞ � j�1j
2 � j�2j2
p.
We now proceed with our first result, which for 2� 2 matrices gives a more detaileddescription of Wð��ðAÞÞ as a function of � than simply (1.4). As a byproduct, it alsocontains an independent proof of Theorem 1.2 for the 2� 2 case.
THEOREM 2.2 For a 2� 2 matrix A and �1, �2 2 ½0, 1�,
Wð��2 ðAÞÞ � Wð��1ðAÞÞ whenever �2 �1
2
�������� � �1 �
1
2
��������: ð2:1Þ
In particular,
Wð��ðAÞÞ ¼ Wð�1��ðAÞÞ for all � 2 ½0, 1�: ð2:2Þ
Proof First, assume that A is singular. It follows from part (iv) of Proposition 1.1 that��ðAÞ is normal for all � 2 ð0, 1Þ. According to part (iv) of Proposition 1.3, for� 2 ð0, 1Þ,Wð��ðAÞÞ is the (possibly degenerate) line segment connecting the eigenvaluesof A while Wð�0ðAÞÞ ¼ Wð�1ðAÞÞ is an ellipse containing this line segment.
Generalized Aluthge transform 165
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We now assume that A is nonsingular and has polar form A¼UR. Due to part (iii) ofProposition 1.1 and part (ii) of Proposition 1.3, we may consider
V�AV ¼ ðV�UV ÞðV�RV Þ
in place of A, where V is an arbitrary unitary matrix. Choosing V in such a way thatV�RV is diagonal, we may therefore without loss of generality suppose that from thebeginning R was of the form
R ¼1 00 a
� �
with a > 0. Since U is an arbitrary 2� 2 unitary matrix, it can be written as
U ¼ei� cos � ei� sin �
ei� sin � ei� cos �
� �
with � 2R and �þ � ¼ �þ � þ . By part (ii) of Proposition 1.1, the generalizedAluthge transform preserves the spectrum, so that the foci of the ellipses Wð��ðAÞÞare the same for all � 2 ½0, 1�. On the other hand,
Trð��ðAÞ���ðAÞÞ ¼ TrðR1��U�R�R�UR1��Þ ¼ TrðU�R2�UR2�2�Þ
and
U�R2�UR2�2� ¼cos2 � þ a2� sin2 � �
� a2�2� sin2 � þ a2 cos2 �
� �,
so that the minor axis of Wð��ðAÞ has the length
lð�Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ a2Þ cos2 � þ ða2� þ a2�2�Þ sin2 � � j�1j
2 � j�2j2
q: ð2:3Þ
The latter as a function of � is nonincreasing on ½0, 1=2� and nondecreasing on½1=2, 1�. g
Note that if A is normal or singular, then all the matrices ��ðAÞ (and therefore theirnumerical ranges) for � 2 ð0, 1Þ are the same. On the other hand, for invertible andnonnormal 2� 2 matrices A the inclusion in (2.1) is strict when j�2 � 1=2j< j�1 � 1=2j.
3. Nesting properties for n>2
A natural question to ask is whether the nesting property of the numerical range underthe generalized Aluthge transform found in Theorem 2.2 for 2� 2 matrices holds formatrices of bigger size. From part (iii) of Proposition 1.3 it follows that it does holdfor all matrices A unitarily similar to direct sums of 1� 1 and 2� 2 blocks, independent
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of the size of A itself. The latter condition is satisfied for example if A belongs to thealgebra generated by two orthogonal projections (see i.e. [8] or [6]).
Nevertheless, there exist 3� 3 matrices for which the nesting property (5) fails.y
To see this, consider for example the matrix
A ¼
1ffiffiffi2
p 512 524288
�1ffiffiffi2
p 512 524288
0 �512ffiffiffi2
p524288
ffiffiffi2
p
2666664
3777775, ð3:1Þ
the polar decomposition (1.1) of which is given by
U ¼
1ffiffiffi2
p1
2
1
2
�1ffiffiffi2
p1
2
1
2
0 �1ffiffiffi2
p1ffiffiffi2
p
266666664
377777775
and
R ¼
1 0 00 1024 00 0 1048576
24
35:
Then
Reð�1=2ðAÞÞ ¼
1ffiffiffi2
p 8ð1�ffiffiffi2
pÞ 256
8ð1�ffiffiffi2
pÞ 512 8192ð1�
ffiffiffi2
pÞ
256 8192ð1�ffiffiffi2
pÞ 524288
ffiffiffi2
p
26664
37775 :¼ X
and
Reð�19=40ðAÞÞ ¼
1ffiffiffi2
p 0 256ffiffiffi2
p
0 512 0
256ffiffiffi2
p0 524288
ffiffiffi2
p
2664
3775 :¼ Y:
A direct computation shows that the largest eigenvalue of Y is
�max ¼1
2ffiffiffi2
p ð1048577þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1099510579201
pÞ < 741460
yFrom Lemma 3.2 it then follows that property (2.2) for these matrices fails as well.
Generalized Aluthge transform 167
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and the characteristic polynomial of X is given by
PXðzÞ ¼ z3 � 512þ1048577ffiffiffi
2p
� �z2 þ 64ð6291462
ffiffiffi2
p� 3138563Þz
þ 67108864ð4ffiffiffi2
p� 9Þ:
Since PXð741460Þ < 0, the matrix X has an eigenvalue larger than the largest eigenvalueof Y. Thus Wð�1=2ðAÞÞ*Wð�19=40ðAÞÞ.
It is of course possible to construct counterexamples of arbitrarily large size by takingthe direct sum of (3.1) with an appropriately sized diagonal matrix whose entries all lieinside the convex hull of �(A).
Although the nesting property fails in general, we will see that the result does hold forcertain 3� 3 matrices. We begin with two supplementary lemmas.
LEMMA 3.1 Given a positive semi-definite 3� 3 matrix H with at most two distincteigenvalues, a 3� 3 unitary matrix U and �> 0, WðHUH�Þ ¼ WðH�UH Þ.
Proof The cases whenH has only one distinct eigenvalue or is singular are trivial, sincethen HUH� ¼ H�UH.
Now let H have two positive distinct eigenvalues. As in the proof of Theorem 2.2we may without loss of generality assume that H is diagonal. Also, using part (i) ofProposition 1.3 we may scale H so that its simple eigenvalue becomes equal 1, that is
H ¼
1 0 00 00 0
24
35
with > 0. Let us partition the matrix U respectively:
U ¼u11 ~u12~u21 U22
� �,
where u11 is a scalar and U22 is a 2� 2 matrix. The singular values of U22 are 1 andjju11 ¼ ! 2 ½0, 1�, following from the fact that U is unitary. By yet another scaling(this time, invoking a unimodular multiple), we may assume without loss of generalitythat u11 ¼ !. Using the singular value decomposition of U22,
U22 ¼ V1! 00 1
� �V2,
we represent U as
U ¼
1 0 000
V1
24
35 ! � �
�
�W
24
35 1 0 0
00
V�1
24
35
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with
W ¼! 00 1
� �V2V1:
Denoting V ¼ V2V1, observe that U is therefore unitarily similar to the matrix
! � �
�
�W
24
35 ¼
! a bc ! 0d 0 1
24
35 1 0 0
00
V
24
35 ð3:2Þ
and the matrices which perform the similarity commute with H. Thus, we can assumethat U was originally of the form (3.2).
The unitarity of U implies that the left multiple in the right hand side of (3.2)actually is
! �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p0
� ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p! 0
0 0 1
24
35
for some � of absolute value 1. Consequently,
U ¼
� 0 0
0 1 0
0 0 1
26664
37775
!ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p0
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p! 0
0 0 1
26664
37775
�� 0 0
0 1 0
0 0 1
26664
37775
1 0 0
0
0V
2664
3775
¼
� 0 0
0 1 0
0 0 1
26664
37775
!ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p0
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p! 0
0 0 1
26664
37775
1 0 0
0
0V
2664
3775
�� 0 0
0 1 0
0 0 1
26664
37775:
It follows that U is unitarily similar to
!ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p0
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p! 0
0 0 1
24
35 1 0 0
00
V
24
35 ð3:3Þ
and the matrices performing this similarity again commute with H. So, it suffices toconsider matrices U of the form (3.3).
Denoting for any �,� 2 R
Y�, � ¼
! �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p0
� �� V
��ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� !2
p
0��
! 00 1
� �� V
24
35,
Generalized Aluthge transform 169
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we then see that
HUH� ¼ Y, � and H�UH ¼ Y� , : ð3:4Þ
To complete the proof, recall that the numerical range of any matrix X is determineduniquely by the homogenous polynomial
LXðu, v,wÞ ¼ detðuReðX Þ þ vImðX Þ � wI Þ;
the fact based on the notion of the associated curve introduced in [11] and then used inparticular in [10].
A direct calculation shows that LY�,�ðu, v,wÞ ¼ LY�,�
ðu, v,wÞ. The result now followsfrom (3.4). g
Our second lemma is a slight generalization of Theorem 1.2.
LEMMA 3.2 Let �2 lie between �0, �1 2 ½0, 1�. Then Wð��2ðAÞÞ is contained in the convexhull of Wð��0 ðAÞÞ and Wð��1ðAÞÞ.
Proof Due to convexity of the sets involved, it suffices to show that any half-planecontaining Wð��0ðAÞÞ and Wð��1ðAÞÞ also contains Wð��2ðAÞÞ. In its turn, property(i) of Proposition 1.1 allows then to consider only half-planes of the formfz : ReðzÞ � �g for �2R. The respective result (up to the equivalence mentioned inpart (vi) of Proposition 1.3) is stated as Lemma 5.2, and its proof is relegated to theappendix. g
We combine the preceding two lemmas to give our result.
THEOREM 3.3 Let A be a 3� 3 matrix with polar form A¼UR and let R have at mosttwo distinct eigenvalues. Then for �1, �2 2 ½0, 1�, j�2 � 1=2j � j�1 � 1=2j implies thatWð��2ðAÞÞ � Wð��1ðAÞÞ; in particular (2.2) holds.
Proof Taking H ¼ R� and � ¼ ð1� �Þ=� in Lemma 3.1 gives
WðR�UR1��Þ ¼ WðR1��UR�Þ,
which is (2.2) for the matrix A under consideration. It remains to invoke Lemma 3.2,using �0 ¼ 1� �1. g
4. The boundary behavior of WðDkðAÞÞ
In this section, we will use the standard notation @ for the boundary operator.Our final result concerns the behavior of the set @Wð��ðAÞÞ as a function of �. In the
case n¼ 2 the situation is rather simple: as follows from formula (2.3), for invertible andnonnormal matrices A the sets @Wð��1ðAÞÞ and @Wð��2 ðAÞÞ are disjoint ifj�1 � 1=2j 6¼ j�2 � 1=2j and coincide otherwise. On the other hand, if A is singularor normal, then @Wð��ðAÞÞ is the line segment connecting the eigenvalues of A,independent of the value of � 2 ð0, 1Þ. The next theorem shows to what extent thisbehavior persists for n� n matrices with n>2. The main role, as we will see, is
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played by the reducing eigenvalues of A. We recall that a reducing eigenvector of A is, bydefinition, a common eigenvector of A and A*. The eigenvalue of A associated with areducing eigenvector is called a reducing eigenvalue.
THEOREM 4.1 Let A be an n� n matrix. Suppose that for some �0 2 ½0, 1=2Þ,�1 2 ð1=2, 1� and �2 lying strictly between �0 and �1, the intersection
Z ¼ @Wð��2ðAÞÞ\
@ conv Wð��0ðAÞÞ,Wð��1 ðAÞÞ
ð4:1Þ
is nonempty. In addition, unless �0 ¼ 0 and �1 ¼ 1, let A be invertible. Then Z consists of( possibly degenerated ) line segments the endpoints of which are reducing eigenvalues of A.Moreover,
Z � Wð��ðAÞÞ for all � 2 ½0, 1� and Z � @Wð��ðAÞÞ for � 2 ½�0, �1�: ð4:2Þ
Proof Due to Lemma 3.2, the set Wð��2 ðAÞÞ (and therefore its boundary) lies inconv Wð��0 ðAÞÞ,Wð��1ðAÞÞ
. Thus,
Z ¼[‘
Wð��2 ðAÞÞ \ ‘
, ð4:3Þ
where the union is taken with respect to all support lines ‘ joint for Wð��jðAÞÞ, j¼ 0, 2.Every intersection in the right-hand side of (4.3) is of course a line segment, possibly
degenerated into a point. We will show that the endpoints of this segment are reducingeigenvalues of A.
Multiplying A by an appropriate scalar if necessary, we may without loss ofgenerality suppose that the support line ‘ in question is vertical and that Wð��j ðAÞÞlies to the left of it for j ¼ 0, 1, 2. Denote by z0, z1 the endpoints of theintersection Wð��2 ðAÞÞ \ ‘. From Lemma 5.3 we concludey that there is at least onereducing eigenvalue of A lying on ‘. Since
�ðAÞ ¼ �ð��ðAÞÞ � Wð��ðAÞÞ for all � 2 ½0, 1�, ð4:4Þ
all such eigenvalues lie in fact in ½z0, z1�. This proves our claim in the degenerate casez0 ¼ z1.
If the segment ½z0, z1� is indeed nondegenerate, let �1, . . . ,�m be all the reducingeigenvalues of A (counting the multiplicities) located on ½z0, z1� and suppose that atleast one of the endpoints (say z0) is not among them. By a unitary similarity put Ain the form
A ¼ BM,
yAs in section 3, we relegate the most technical part of the proof to the appendix.
Generalized Aluthge transform 171
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where M is the m�m diagonal matrix with �j on the diagonal, and B does not havereducing eigenvalues lying on ‘. Then
��ðAÞ ¼ ��ðBÞ M,
so that by part (iii) of Proposition 1.3
Wð��ðAÞÞ ¼ convðWð��ðBÞÞ,�1, . . . ,�mÞ, � 2 ½0, 1�:
In particular, z0 2 Wð��2 ðBÞÞ, while Wð��jðBÞÞ lies to the left of ‘ for j ¼ 0, 1, 2.Applying Lemma 5.3 to the matrix B we see that it must have reducing eigenvalueson ‘ – a contradiction with the way B was constructed.
This proves the description of the set Z from the statement of the theorem. The firstinclusion in (4.2) now follows from (4.4). To derive the second from the first, oneneeds only to invoke Lemma 3.2, with �2 in its statement changed to an arbitrary� 2 ð�0, �1Þ. g
The statement of Theorem 4.1 takes its simplest (and probably the most useful) formwhen �0 ¼ 0, �1 ¼ 1.
COROLLARY 4.2 Let A be an n� n matrix. Then the intersection
@Wð��ðAÞÞ \ @WðAÞ
does not depend on � 2 ð0, 1Þ. More specifically, it is either empty (that is, Wð��ðAÞÞ liesstrictly inside W(A)), or consists of ( possibly degenerated ) line segments, the endpointsof which are reducing eigenvalues of A.
In particular, the boundaries of W(A) and Wð��ðAÞÞ are disjoint for � 2 ð0, 1Þ in caseof unitarily irreducible 3� 3 matrices A. Indeed, such matrices do not have reducingeigenvalues.
We conclude by an example showing that invertibility of A cannot be dropped in thestatement of Theorem 4.1. To this end, let A¼ Jn, the n� n Jordan block with the zeroeigenvalue. A direct computation shows that ��ðAÞ ¼ 0 Jn�1 for all � 2 ð0, 1Þ. Fromhere it follows (see e.g. [12]) that Wð��ðAÞÞ is the disk centered at the origin and theradius cos=n for all � 2 ð0, 1Þ. The set Z defined by (4.1) is therefore the circlefz : jzj ¼ cos=ng for any choice of �j 2 ð0, 1Þ, j ¼ 0, 1, 2. Nevertheless, A has no redu-cing eigenvalues.
Observe that @Wð��ðAÞÞ \ @WðAÞ ¼ 6 0 for all � 2 ð0, 1Þ, which agrees withCorollary 4.2.
Acknowledgements
The work on this article was started in the summer of 2005 during the REU program atthe College of William and Mary and was then supported by NSF GrantDMS-0353510. During its completion both authors were supported in part by NSFGrant DMS-0456625. The results were presented at the 8th Workshop on NumericalRanges and Numerical Radii, Universitat Bremen, Germany, July 15–17, 2006.
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References
[1] Aluthge, A., 1999, On p-hyponormal operators for 0 < p < 1. Integral Equations and Operator Theory,13(3), 307–315.
[2] Aluthge, A., 1996, Some generalized theorems on p-hyponormal operators. Integral Equations andOperator Theory, 24(4), 497–501.
[3] Ando, T., 2004, Aluthge transforms and the convex hull of the eigenvalues of a matrix. Linear MultilinearAlgebra, 52(3–4), 281–292.
[4] Ando, T. and Yamazaki, T., 2003, The iterated Aluthge transforms of a 2-by-2 matrix converge. LinearAlgebra and its Applications, 375, 299–309.
[5] Antezana, J., Massey, P. and Stojanoff, D., 2005, �-Aluthge transforms and Schatten ideals. LinearAlgebra and its Applications, 405, 177–199.
[6] Giles, R. and Kummer, H., 1971, A matrix representation of a pair of projections in Hilbert space.Canadian Mathematical Bulletin, 14(1), 35–44.
[7] Gustafson, K.E. and Rao, D.K.M., 1997, Numerical Range. The Field of Values of Linear Operators andMatrices (New York: Springer).
[8] Halmos, P.R., 1969, Two subspaces. Transactions of American Mathematical Society, 141, 381–389.[9] Jung, I.B., Ko, E. and Pearcy, C., 2000, Aluthge transforms of operators. Integral Equations and
Operator Theory, 37(4), 437–448.[10] Keeler, D., Rodman, L. and Spitkovsky, I., 1997, The numerical range of 3� 3 matrices. Linear Algebra
and its Applications, 252, 115–139.[11] Kippenhahn, R., 1951, Uber den Wertevorrat einer Matrix. Mathematische Nachrichten, 6, 193–228.[12] Marcus, A. and Shure, B.N., 1979, The numerical range of certain 0, 1-matrices. Linear and Multilinear
Algebra, 7, 111–120.[13] Okubo, K., 2003, On weakly unitarily invariant norm and the Aluthge transformation. Linear Algebra
and its Applications, 371, 369–375.[14] Okubo, K., 2006, On weakly unitarily invariant norm and the �-Aluthge transformation for invertible
operator. Linear Algebra and its Applications, 419, 48–52.[15] Yamazaki, T., 2002, On numerical range of the Aluthge transformation. Linear Algebra and its
Applications, 341, 111–117.
Appendix
This auxiliary section contains two lemmas. Their statements and proofs are slightvariations of a beautiful argument used by Ando in [3] when proving his Theorem 3on the characterization of convexoid matrices. Since our setting is formally different(we consider the generalized Aluthge transform, as opposed to the regular one consid-ered in [3]), and since we need some properties not stated in [3] explicitly (such as thereducing property of the eigenvalue the existence of which is established inLemma 5.3), we include the complete proofs for the sake of self-containment.
Prompted by the referee’s question, we also note that our version of Lemma 5.3implies the following result, slightly generalzing the above mentioned Ando’s theorem.
THEOREM 5.1 An n� n matrix A is convexoid (that is, W(A) coincides with the convexhull of the spectrum of A) if and only if WðAÞ ¼ Wð��ðAÞÞ for some (equivalently: all )� 2 ð0, 1Þ.
LEMMA 5.2 Let for some �0, �1 2 ½0, 1� and �2R:
Re��0ðAÞ � �I and Re��1ðAÞ � �I: ð5:1Þ
Then also
Re��2ðAÞ � �I
for any �2 lying between �0 and �1.
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Proof Represent the multiple R from the polar decomposition (1.1) in the formR ¼ R1 0 with R1 nonsingular, and define powers R as Rz ¼ Rz
1 0, using realvalues of logarithms for the eigenvalues of R1. Then
�ðzÞ ¼ Rð1=2Þ�zURð1=2Þþz ð5:2Þ
is an entire matrix function. Without loss of generality, let us suppose that �0 < �1, andconsider � on the strip fz : �0 � Re z � �1g.
For any � 2 ½0, 1� and t 2 R,
�1
2� �þ it
� �x, x
� �¼ hR�UR1��ðRitxÞ,Ritxi ¼ h��ðAÞR
itx,Ritxi:
Noting that kRitxk � kxk we thus have Re�ðzÞ � �I on the lines Re z ¼ 12 � �0 and
Re z ¼ 12 � �1. It then follows by the three line theorem for analytic functions that
Re�ðzÞ � �I for all z 2 S: ð5:3Þ
In particular, Re��2 ðAÞ � �I. g
LEMMA 5.3 Let �0ð<12 <Þ�1 be some points in ½0, 1� for which (5.1) holds, and in addition
Wð��2 ðAÞÞ \ fz : Re z ¼ �g 6¼ 6 0 ð5:4Þ
for some �2 2 ð�0, �1Þ. Suppose that also either (i) �0 ¼ 0, �1 ¼ 1 or (ii) A is invertible.Then there exists a reducing eigenvalue of A lying on the line fz:Re z ¼ �g.
Proof We will be using the notation introduced in the proof of Lemma 5.2. Since��2 ðAÞ ¼ � 1=2� �2ð Þ, condition (5.4) implies that there exists a vector u such thatkuk ¼ 1 and
Re �1
2� �2
� �� �u, u
� �¼ �: ð5:5Þ
Let us fix this u and consider the scalar valued analytic function � given by
�ðzÞ ¼ h�ðzÞu, ui: ð5:6Þ
From condition (5.3) proved in Lemma 5.2 it follows that
Re�ðzÞ � � on S: ð5:7Þ
On the other hand,
Re �1
2� �2
� �� �¼ �
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due to (5.5). The maximum modulus principle then gives
Re�ðzÞ ¼ � for all z 2 S: ð5:8Þ
Equations (5.3) and (5.8) then yield
�I�Re�ðzÞ½ �u ¼ 0 for all z 2 S, ð5:9Þ
in particular
�I�Re�ð0Þ½ �u ¼ 0: ð5:10Þ
Denoting by Q the orthogonal projection onto the range L of R, observe that
Re�ð0Þ ¼ R1=2ðReU ÞR1=2 ¼ QR1=2ðReU ÞR1=2Q
and rewrite (5.10) in the form
�ðI�QÞu ¼ 0, Qu 2 M, ð5:11Þ
where
M ¼ ker �I�Re�ð0Þ½ �\
L:
If Qu¼ 0 then the equality in (5.11) implies that also �¼ 0. Since u 2 kerR ¼ kerA,zero is an eigenvalue of A. Moreover, since A in this case is singular, we musthave �0 ¼ 0. This, due to (5.1), means that 0 lies on the boundary of W(A) and istherefore a reducing eigenvalue.
It remains to consider the case when Qu 6¼ 0. The subspace M is then nontrivial.Moreover, substituting in the definition (5.6) of � the vector u by any unit vectorin M we see that �ð0Þ ¼ 0 while (5.7) still holds. Repeating the maximum modulusprinciple argument in this setting we again arrive at (5.9). Taking this time z¼ it,t 2 R, we conclude
�I�Re�ð0Þ½ �Ritu ¼ 0:
In other words, the subspaceM is invariant under Rit, and therefore under R and, moregenerally, any polynomial f of R. Thus for any x 2 M we have
R1=2ðReU ÞR1=2fðRÞx ¼ �fðRÞx:
Approximating R�1=21 0 by polynomials of R it follows from here that for x 2 M
R1=2ðReU ÞQx ¼ �ðR�1=21 0Þx, ð5:12Þ
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or equivalently QðReU ÞQx ¼ �ðR�11 0Þx. Since M is invariant under R�1
1 0 thisshows M is invariant under QðReUÞQ as well.
Also, (5.12) implies that on M
RQðReU ÞQx ¼ QðReU ÞQRx ¼ �x: ð5:13Þ
In other words, the hermitian mappings R and QðReU ÞQ commute on M andtherefore have a common eigenvector, say v. Then
Qv ¼ v, Rv ¼ �v, and QðReU Þv ¼ v cos � ð5:14Þ
for some �>0 and �2R such that � cos � ¼ �.We now claim that either of conditions (i) or (ii) guarantees that
QðReU Þv ¼ ðReU Þv: ð5:15Þ
If (ii) holds, then Q¼ I which makes (5.15) trivial. In its turn, condition (i) means that
Re ðURÞ � �I, Re ðRU Þ � �I:
But, according to (5.14),
hð�I�Re ðURÞÞv, vi ¼ hð�I�Re ðRU ÞÞv, vi ¼ 0:
Consequently,
½�I�Re ðURÞ�v ¼ 0 and ½�I�Re ðRU Þ�v ¼ 0:
Using (5.14) again, the latter two equalities can be rewritten as
�ðI�QÞðReU Þvþ ið�I� RÞðImU Þv ¼ 0 ð5:16Þ
and
�ðI�QÞðReU Þv� ið�I� RÞðImU Þv ¼ 0, ð5:17Þ
respectively. Since �>0, (5.16) and (5.17) imply (5.15).Consequently, v is an eigenvector of ReU corresponding to the eigenvalue cos �.
Since U is unitary, v must then be of the form v ¼ vþ þ v� where
Uvþ ¼ ei�vþ, Uv� ¼ e�i�v�: ð5:18Þ
In particular,
ðImU Þv ¼ sin �ðvþ � v�Þ: ð5:19Þ
If sin � ¼ 0, then it follows that
Uv ¼ ðReU Þv ¼ cos �v:
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This makes v a common eigenvector of U and R, and thus a reducing eigenvectorof A¼UR. The respective eigenvalue is � cos � ¼ �, as we wanted.
Finally, let sin � 6¼ 0. Then (5.16) or (5.17), when compared with (5.19), implies
ð�I� RÞðvþ � v�Þ ¼ 0:
Combined with the middle equality in (5.14), this means that
ð�I� RÞvþ ¼ ð�I� RÞv� ¼ 0:
At least one of the vectors v is different from zero. It is therefore a commoneigenvector of U and R, and consequently a reducing eigenvector of A. The respectiveeigenvalue is either �ei� or �e�i�, with the real part equal �. g
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