In this paper we establish several equivalent characterizations, some of which provide a natural distinction between boundary points and interior points, and hence should prove useful for computations. Additionally, we use one of these equivalent definitions to prove that the polynomial numerical hull of any fixed degree $k$ for a Toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite-dimensional Toeplitz operator, as the matrix goes to infinity

Burke, James

Greenbaum, Anne

Danish

Oxford University Research Archive

Some Equivalent Charaterizations of thePolynomial Numerial Hull of Degree kJames Burke Anne GreenbaumyDeember 16, 2004AbstratThe polynomial numerial hull of degree k for a square matrix orbounded linear operator A on a Hilbert spae is dened asHk(A) = fz 2 C : kp(A)k  jp(z)j 8p 2 Pkg;where Pk denotes the set of polynomials of degree k or less. An equiv-alent denition isHk(A) = f 2 C : min1;:::;k kI   kXj=1 j(A  I)jk = 1g:In this paper we establish several equivalent haraterizations, someof whih provide a natural distintion between boundary points andinterior points, and hene should prove useful for omputations. Ad-ditionally, we use one of these equivalent denitions to prove that thepolynomial numerial hull of any xed degree k for a Toeplitz matrixwhose symbol is pieewise ontinuous approahes all or most of thatof the innite-dimensional Toeplitz operator, as the matrix size goesto innity.University of Washington, Mathematis Dept., Box 354350, Seattle, WA 98195, USA.yUniversity of Washington, Mathematis Dept., Box 354350, Seattle, WA 98195, USA.This work was supported in part by NSF grant DMS-0208353.11 IntrodutionThe polynomial numerial hull of degree k for a square matrix or boundedlinear operator A on a Banah spae is dened asHk(A) := fz 2 C : kp(A)k  jp(z)j 8p 2 Pkg; (1)where Pk denotes the set of polynomials of degree k or less [8, 9, 4℄. The setsHk(A) are nonempty and ompat, and H1(A) is equal to the losed onvexhull of the eld of values of A [8℄. In this paper, we will restrit attention tooperators on a Hilbert spae, where the eld of values is a onvex set, andwe dene F(A) to be the losure of that set:F(A) := l (fhAq; qi : kqk = 1g) : (2)In a nite-dimensional spae, the eld of values is losed, but this is notneessarily so for innite-dimensional spaes.An equivalent haraterization of the polynomial numerial hull of degreek is [4℄ Hk(A) = f 2 C : min1;:::;k kI   kXj=1 j(A  I)jk = 1g: (3)In other words,  2 Hk(A) if and only if a solution to the minimizationproblem min1;:::;k kI  Pkj=1 j(A  I)jk is 1 = : : : = k = 0. In this paperwe establish several equivalent haraterizations, some of whih provide anatural distintion between boundary points and interior points, and heneshould prove useful for omputations. Additionally, we use one of theseequivalent denitions to prove that the polynomial numerial hull of anyxed degree k for a Toeplitz matrix whose symbol is pieewise ontinuousapproahes all or most of that of the innite-dimensional Toeplitz operator, asthe matrix size goes to innity. More preisely, we show that the polynomialnumerial hull of degree k for the Toeplitz operator ontains the uniform andpartial limits as N ! 1 of that of the N by N Toeplitz matrix, whih inturn ontain the losure of the interior of that of the Toeplitz operator.22 Equivalent CharaterizationsIt follows from denition (3) and the identity H1(A) = F(A), that for anybounded linear operator B,min2C kI   Bk < 1 if and only if 0 =2 F(B):It also follows from (3) that a point  lies outside Hk(A) if and only if thereexist salars 1; : : : ; k suh thatkI   kXj=1 j(A  I)jk < 1;or, equivalently, suh thatmin2C kI   ( kXj=1 j(A  I)j)k < 1:Hene we have  =2 Hk(A) if and only if9 1; : : : ; k suh that 0 =2 F 0 kXj=1 j(A  I)j1A : (4)An equivalent statement is 2 Hk(A) if and only if81; : : : ; k; 0 2 F 0 kXj=1 j(A  I)j1A : (5)The following theorem gives a riterion for distinguishing boundary pointsof Hk(A) from interior points.Theorem 1. A point  lies on the boundary of Hk(A) if and only if: for all1; : : : ; k 0 2 F 0 kXj=1 j(A  I)j1A ; (6)3and for some ̂1; : : : ; ̂k, not all zero,0 2 F 0 kXj=1 ̂j(A  I)j1A ; (7)where  denotes the boundary.Proof: Condition (6) was already shown to be equivalent to the statement 2 Hk(A); hene it holds for  2 Hk(A) sine the set is losed.To show that  2 Hk(A) implies (7), let  be a point on the boundary ofHk(A), and let fmg1m=0 be a sequene of points outside Hk(A) and onverg-ing to . For eah point m, it follows from (4) that there exist oeÆients1;m; : : : ; k;m suh that 0 =2 F 0 kXj=1 j;m(A  mI)j1A :We an take these oeÆients to satisfy, say, Pkj=1 jj;mj2 = 1, sine multiply-ing an operator by a nonzero salar just multiplies its eld of values by thatsalar and does not aet whether or not the losure of the eld of valuesontains the origin. Sine the vetors (1;m; : : : ; k;m; m) are bounded, thereexists a onvergent subsequene, f(1;m`; : : : ; k;m`; m`)g1̀=1, onverging to,say, (̂1; : : : ; ̂k; ). Sine the eld of values is a ontinuous funtion of thelinear operator [6℄ and sine0 =2 F( kXj=1 j;m`(A  m`I)j)! F( kXj=1 ̂j(A  I)j) 3 0 (by (5));it follows that 0 2 F(Pkj=1 ̂j(A  I)j).Conversely, suppose  is a point in the interior ofHk(A) and suppose that0 2 F 0 kXj=1 j(A  I)j1A ;for ertain salars 1; : : : ; k. We will show that eah j must be 0.Sine Hk(A) ontains a disk of radius, say, r > 0 about , it follows from(5) that for any salars d1; : : : ; dk, any Æ 2 [0; r), and any  2 [0; 2),0 2 F 0 kXj=1 dj(A  (   Æei)I)j1A : (8)4Expanding using the binomial formula, we ndkXj=1 dj(A  (   Æei)I)j = kXj=1 dj jX̀=0 j̀ ! Æ`ei`(A  I)j `= kXj=1 dj(A  I)j + kXj=1 dj j 1X̀=1 j̀ ! Æ`ei`(A  I)j `+0 kXj=1 djÆjeij1A I= kXj=10dj + k jX̀=1 dj+` j + `` ! Æ`ei`1A (A  I)j+0 kXj=1 djÆjeij1A I:Suppose dk = k and dj, j = k   1; :::; 1 are determined bydj = j   k jX̀=1 dj+` j + `` ! Æ`ei`: (9)Then kXj=1 dj(A  (   Æei)I)j = kXj=1 j(A  I)j + 0 kXj=1 djÆjeij1A I; (10)and the eld of values of this matrix is that of Pkj=1 j(A   I)j shiftedby Pkj=1 djÆjeij. To express Pkj=1 djÆjeij in terms of the j's, note thatif Pkj=1 j(A   I)j = Pkj=1 j(A   (   Æei)I   ÆeiI)j is expanded usingthe binomial formula, then the expansion involves powers from 1 to k ofA   (   Æei)I plus the term (Pkj=1 j( 1)jÆjeij)I. Comparing this with(10), it follows that kXj=1 djÆjeij = kXj=1 j( 1)j+1Æjeij: (11)If the origin is on the boundary of F(Pkj=1 j(A   I)j), then there is aline through the origin that separates the eld of values from a half-plane.5Suppose 1 = : : : = ` 1 = 0 but ` 6= 0. Choose  so that ei`( 1)`+1`lies in the half-plane ontaining F(Pkj=1 j(A   I)j) and is orthogonal tothe separating line. Then if the eld of values is shifted in the diretion ofei`( 1)`+1` its losure will exlude the origin. By hoosing Æ > 0 suÆientlysmall, one an make the shift term in (11) arbitrarily lose to `( 1)`+1Æ`ei`,and so one an exlude the origin from the losure of the eld of values ofPkj=1 dj(A  (  Æei)I)j, but this ontradits (8). Therefore eah oeÆientj must be 0. 2Theorem 1 and (5) imply also that 2 Int(Hk(A)) if and only if81; : : : ; k; not all zero, 0 2 Int0F 0 kXj=1 j(A  I)j1A1A ; (12)where Int denotes the interior.If the origin lies outside the losure of the eld of values of an operatorB, then F(B) an be rotated into the right half-plane by multiplying B bya ertain salar ei. Sine the real part of the losure of the eld of values,ReF(eiB) := fRe z : z 2 F(eiB)g, is the onvex hull of the spetrum ofthe Hermitian part of eiB, H(eiB) := (eiB + e iB)=2, [6, pp. 9,21℄, itfollows that H(eiB) is positive denite. Therefore an equivalent statementof (4) is  =2 Hk(A) if and only if9 1; : : : ; k suh that H( kXj=1 j(A  I)j) is positive denite, (13)and (5) beomes  2 Hk(A) if and only if81; : : : ; k; H( kXj=1 j(A  I)j) has a nonpositive eigenvalue. (14)Theorem 1 an be stated as 2 Hk(A) if and only if8 1; : : : ; k; min0H( kXj=1 j(A  I)j)1A  0; and69 ̂1; : : : ; ̂k; not all 0, suh that min0H( kXj=1 ̂j(A  I)j)1A = 0; (15)where min denotes the smallest eigenvalue. In other words, if one hooses1; : : : ; k with, say, Pkj=1 jjj2 = 1, to maximize min(H(Pkj=1 j(A  I)j)),then this minimum eigenvalue will be positive if  =2 Hk(A), zero if  2Hk(A), and negative if  2 Int(Hk(A)).We note one more equivalent denition. In [2℄, Davies introdues the setNumk(A) := fz 2 C : p(z) 2 F(p(A)) 8p 2 Pkg:From (5), and the fat that adding a salar times the identity to an operatorshifts its eld of values by that salar, we an write 2 Hk(A) if and only if8 0; 1; : : : ; k; 0 2 F 00I + kXj=1 j(A  I)j1A : (16)Sine any polynomial p of degree k or less in A an be written in the form0I+Pkj=1 j(A I)j for ertain oeÆients 0; 1; : : : ; k, and then p() = 0,equivalene (16) implies 2 Hk(A) () 8 p 2 Pk; p() 2 F(p(A));i.e., Hk(A) = Numk(A). It is interesting to note that although Hk(A) andNumk(A) are the same for bounded operators, only the latter denitionmakes sense for unbounded operators.3 Polynomial Numerial Hulls of Toeplitz Ma-tries and OperatorsLet T denote an innite-dimensional Toeplitz operator:T = 0BBBB a0 a 1 a 2 : : :a1 a0 a 1 : : :a2 a1 a0 : : :... ... ... . . . 1CCCCA : (17)7The symbol of T evaluated at a point z is dened asa(z) := a0 + 1X̀=1 a`z` + 1X̀=1 a `z `: (18)We will assume throughout that a(z) 2 L1(U), where U := fz : jzj = 1gdenotes the unit irle, so that the matrix in (17) orresponds to a boundedlinear operator on `2 (see, e.g., [1℄).The polynomial numerial hull of degree 1 for T , i.e., the losure of theeld of values, is known [7℄: H1(T ) = o (a(U)) ; (19)where o() denotes the onvex hull.Theorem 2. The polynomial numerial hull of degree k  1 for T satiseso (a(U))  Hk(T )  pok (a(U)) : (20)Here pok() denotes the polynomially onvex hull of degree k, dened forany ompat set S  C aspok(S) := fz 2 C : jp(z)j  max2S jp()j 8p 2 Pkg:Proof: The left inlusion follows from (19) and the fat that Hk(T ) H1(T ). To establish the right inlusion, let p(T ) be any polynomial in T . Bythe spetral mapping theorem, (p(T )) = p((T )), where () denotes thespetrum. By the Hartman-Wintner theorem (see, e.g., [1, Theorem 1.25,p. 27℄), the essential spetrum of T ontains the range of its symbol. Henekp(T )k  sup2(p(T )) jj  supjzj=1 jp(a(z))j :The result then follows from denition (1) of Hk(T ). 2In words, the polynomial numerial hull of degree k for an innite-dimensionalToeplitz operator lies somewhere between the polynomially onvex hull of de-gree k and the ordinary onvex hull of the image, under the symbol, of theunit irle. 8Let TN be the N by N Toeplitz matrix onsisting of the upper left blokof the operator in (17):TN = 0BBBBBBB a0 a 1 a 2 : : : a (N 1)a1 a0 a 1 : : : a (N 2)a2 a1 a0 : : : a (N 3)... ... ... . . . ...aN 1 aN 2 aN 3 : : : a01CCCCCCCA : (21)It is known that if T is banded or, more generally, if a(z) is pieewise ontin-uous on the unit irle, then the polynomial numerial hull of degree 1 (i.e.,the eld of values) of TN approahes that of T as N ! 1 [3, 5, 10℄. More-over, if p(TN) is any polynomial in TN and if a(z) is pieewise ontinuous onthe unit irle, then it follows from results in [10, Theorem 2℄ thatp-limN!1F(p(TN)) = u-limN!1F(p(TN)) = F(p(T )): (22)Here u-limN!1SN , for a sequene of sets SN , denotes the set of all limits of se-quenes of points fsN 2 SNg1N=1, while p-limN!1SN denotes all limits of sub-sequenes fsN` 2 SN`g1̀=1, where (N1; N2; : : :) is a subsequene of (1; 2; : : :).Clearly u-limN!1SN  p-limN!1SN . Note that if  2 Int(F(p(T ))), then(22) implies that for N large enough,  2 Int (F(p(TN))), sine these sets areonvex and they ontain sequenes of points approahing points on a irleabout  in Int(F(p(T ))). [See, for example, [11, Prop. 4.15℄.℄Theorem 3. Assume a(z) in (18) is pieewise ontinuous on the unit irle.For any xed degree k  1,Hk(T )  p-limN!1Hk(TN)  u-limN!1Hk(TN)  l (Int (Hk(T ))) ; (23)where l() denotes the losure.Proof: First suppose N` 2 Hk(TN`) is a onvergent partial sequenewith N` !  as ` ! 1. Then it follows from (5) that for all 1; : : : ; k,0 2 F(Pkj=1 j(TN` N`I)j). Sine the eld of values is a ontinuous funtionof the operator, it follows that for eah 1; : : : ; k, there is a sequene of pointssN` 2 F(Pkj=1 j(TN`   I)j) suh that sN` ! 0 as ` ! 1. By (22), thisimplies that 0 2 F(Pkj=1 j(T   I)j), and hene by (5),  2 Hk(T ). This9establishes the rst inlusion in (23). The seond inlusion is lear from thedenitions of u-lim and p-lim.Now let  be any point in Int(Hk(T )). Then we know from (12) thatfor all 1; : : : ; k with, say, Pkj=1 jjj2 = 1, 0 2 Int F Pkj=1 j(T   I)j.It follows from (22) that there exists n  n(1; : : : ; k) suh that for allN  n, 0 2 Int(F(Pkj=1 j(TN   I)j)). Sine the oeÆients ome from aompat set and sine the eld of values is a ontinuous funtion of 1; : : : ; k,there exist oeÆients ̂1; : : : ; ̂k for whih the required minimum n-value,n̂  n(̂1; : : : ; ̂k), is maximal and nite. Therefore, for N  n̂ and for all1; : : : ; k withPkj=1 jjj2 = 1, 0 2 Int(F(Pkj=1 j(TN I)j)). Hene by (12), 2 Int(Hk(TN )). This shows that every interior point of Hk(T ) eventuallylies in Hk(TN ) and hene that every limit of interior points of Hk(T ) is alimit of points in Hk(TN). This proves the third inlusion in (23). 2The preeding theorems leave open the possibility that Hk(T ) ontainsisolated points or urves between the region enlosed by a(U) and o(a(U)),and, in this ase, it is not known if there is a sequene or partial sequeneof points in Hk(TN ) onverging to these points. In the simplest ase, wherethe region enlosed by a(U) is a onvex set, Theorems 2 and 3 imply thatp-limN!1Hk(TN ) = u-limN!1Hk(TN ) = Hk(T ) = o(a(U)).Finally, note that Theorem 3 makes no diret use of Toeplitz properties;it uses only relation (22) and properties of the polynomial numerial hull.Hene for any sequene of matries or operators AN satisfyingp-limN!1F(p(AN)) = u-limN!1F(p(AN)) = F(p(A));for all polynomials p, the analogue of relation (23) will hold, namely,Hk(A)  p-limN!1Hk(AN )  u-limN!1Hk(AN)  l (Int (Hk(A))) ;for any xed k  1.Aknowledgment: The authors thank Nik Trefethen for helpful ommentson a draft of this paper.Referenes[1℄ A. Botther and B. Silbermann, Introdution to Large Trunated ToeplitzMatries, Springer, 1999. 10[2℄ E. B. Davies, Spetral bounds using higher order numerial ranges,manusript to appear, 2004.[3℄ M. Eiermann, Fields of values and iterative methods, Lin. Alg. Appl.180 (1993), pp. 167{197.[4℄ A. Greenbaum, Generalizations of the eld of values useful in the studyof polynomial funtions of a matrix, Lin. Alg. Appl. 347 (2002), pp. 233{249.[5℄ A. Greenbaum, Some theoretial results derived from polynomial numer-ial hulls of Jordan bloks, ETNA 18 (2004), pp. 81{90.[6℄ R. A. Horn and C. R. Johnson, Topis in Matrix Analysis, CambridgeUniversity Press, 1991.[7℄ E. M. Klein, The numerial range of a Toeplitz operator, Pro. Amer.Math. So. 35 (1972), pp. 101{103.[8℄ O. Nevanlinna, Convergene of Iterations for Linear Equations,Birkhauser, Basel, 1993.[9℄ O. Nevanlinna, Hessenberg matries in Krylov subspaes and the om-putation of the spetrum, Numer. Funt. Anal. and Optimiz., 16 (1995),pp. 443{473.[10℄ S. Roh, Numerial ranges of large Toeplitz matries, Lin. Alg. Appl.282 (1998), pp. 185{198.[11℄ R. T. Rokafellar and J. B. Wets, Variational Analysis, Springer, 1998.11

English

Some Equivalent Characterizations of the Polynomial Numerical Hull of Degree $k$

Mathematical Institute Eprints Archive

Some Equivalent Charaterizations of the
Polynomial Numerial Hull of Degree k
James Burke

Anne Greenbaum
y
Deember 16, 2004
Abstrat
The polynomial numerial hull of degree k for a square matrix or
bounded linear operator A on a Hilbert spae is dened as
H
k
(A) = fz 2 C : kp(A)k  jp(z)j 8p 2 P
k
g;
where P
k
denotes the set of polynomials of degree k or less. An equiv-
alent denition is
H
k
(A) = f 2 C : min

1
;:::;
k
kI  
k
X
j=1

j
(A  I)
j
k = 1g:
In this paper we establish several equivalent haraterizations, some
of whih provide a natural distintion between boundary points and
interior points, and hene should prove useful for omputations. Ad-
ditionally, we use one of these equivalent denitions to prove that the
polynomial numerial hull of any xed degree k for a Toeplitz matrix
whose symbol is pieewise ontinuous approahes all or most of that
of the innite-dimensional Toeplitz operator, as the matrix size goes
to innity.

University of Washington, Mathematis Dept., Box 354350, Seattle, WA 98195, USA.
y
University of Washington, Mathematis Dept., Box 354350, Seattle, WA 98195, USA.
This work was supported in part by NSF grant DMS-0208353.
1
1 Introdution
The polynomial numerial hull of degree k for a square matrix or bounded
linear operator A on a Banah spae is dened as
H
k
(A) := fz 2 C : kp(A)k  jp(z)j 8p 2 P
k
g; (1)
where P
k
denotes the set of polynomials of degree k or less [8, 9, 4℄. The sets
H
k
(A) are nonempty and ompat, and H
1
(A) is equal to the losed onvex
hull of the eld of values of A [8℄. In this paper, we will restrit attention to
operators on a Hilbert spae, where the eld of values is a onvex set, and
we dene F(A) to be the losure of that set:
F(A) := l (fhAq; qi : kqk = 1g) : (2)
In a nite-dimensional spae, the eld of values is losed, but this is not
neessarily so for innite-dimensional spaes.
An equivalent haraterization of the polynomial numerial hull of degree
k is [4℄
H
k
(A) = f 2 C : min

1
;:::;
k
kI  
k
X
j=1

j
(A  I)
j
k = 1g: (3)
In other words,  2 H
k
(A) if and only if a solution to the minimization
problem min

1
;:::;
k
kI  
P
k
j=1

j
(A  I)
j
k is 
1
= : : : = 
k
= 0. In this paper
we establish several equivalent haraterizations, some of whih provide a
natural distintion between boundary points and interior points, and hene
should prove useful for omputations. Additionally, we use one of these
equivalent denitions to prove that the polynomial numerial hull of any
xed degree k for a Toeplitz matrix whose symbol is pieewise ontinuous
approahes all or most of that of the innite-dimensional Toeplitz operator, as
the matrix size goes to innity. More preisely, we show that the polynomial
numerial hull of degree k for the Toeplitz operator ontains the uniform and
partial limits as N ! 1 of that of the N by N Toeplitz matrix, whih in
turn ontain the losure of the interior of that of the Toeplitz operator.
2
2 Equivalent Charaterizations
It follows from denition (3) and the identity H
1
(A) = F(A), that for any
bounded linear operator B,
min
2C
kI   Bk < 1 if and only if 0 =2 F(B):
It also follows from (3) that a point  lies outside H
k
(A) if and only if there
exist salars 
1
; : : : ; 
k
suh that
kI  
k
X
j=1

j
(A  I)
j
k < 1;
or, equivalently, suh that
min
2C
kI   (
k
X
j=1

j
(A  I)
j
)k < 1:
Hene we have
 =2 H
k
(A) if and only if
9 
1
; : : : ; 
k
suh that 0 =2 F
0

k
X
j=1

j
(A  I)
j
1
A
: (4)
An equivalent statement is
 2 H
k
(A) if and only if
8
1
; : : : ; 
k
; 0 2 F
0

k
X
j=1

j
(A  I)
j
1
A
: (5)
The following theorem gives a riterion for distinguishing boundary points
of H
k
(A) from interior points.
Theorem 1. A point  lies on the boundary of H
k
(A) if and only if: for all

1
; : : : ; 
k
0 2 F
0

k
X
j=1

j
(A  I)
j
1
A
; (6)
3
and for some ^
1
; : : : ; ^
k
, not all zero,
0 2 F
0

k
X
j=1
^
j
(A  I)
j
1
A
; (7)
where  denotes the boundary.
Proof: Condition (6) was already shown to be equivalent to the statement
 2 H
k
(A); hene it holds for  2 H
k
(A) sine the set is losed.
To show that  2 H
k
(A) implies (7), let  be a point on the boundary of
H
k
(A), and let f
m
g
1
m=0
be a sequene of points outside H
k
(A) and onverg-
ing to . For eah point 
m
, it follows from (4) that there exist oeÆients

1;m
; : : : ; 
k;m
suh that
0 =2 F
0

k
X
j=1

j;m
(A  
m
I)
j
1
A
:
We an take these oeÆients to satisfy, say,
P
k
j=1
j
j;m
j
2
= 1, sine multiply-
ing an operator by a nonzero salar just multiplies its eld of values by that
salar and does not aet whether or not the losure of the eld of values
ontains the origin. Sine the vetors (
1;m
; : : : ; 
k;m
; 
m
) are bounded, there
exists a onvergent subsequene, f(
1;m
`
; : : : ; 
k;m
`
; 
m
`
)g
1
`=1
, onverging to,
say, (^
1
; : : : ; ^
k
; ). Sine the eld of values is a ontinuous funtion of the
linear operator [6℄ and sine
0 =2 F(
k
X
j=1

j;m
`
(A  
m
`
I)
j
)! F(
k
X
j=1
^
j
(A  I)
j
) 3 0 (by (5));
it follows that 0 2 F(
P
k
j=1
^
j
(A  I)
j
).
Conversely, suppose  is a point in the interior ofH
k
(A) and suppose that
0 2 F
0

k
X
j=1

j
(A  I)
j
1
A
;
for ertain salars 
1
; : : : ; 
k
. We will show that eah 
j
must be 0.
Sine H
k
(A) ontains a disk of radius, say, r > 0 about , it follows from
(5) that for any salars d
1
; : : : ; d
k
, any Æ 2 [0; r), and any  2 [0; 2),
0 2 F
0

k
X
j=1
d
j
(A  (   Æe
i
)I)
j
1
A
: (8)
4
Expanding using the binomial formula, we nd
k
X
j=1
d
j
(A  (   Æe
i
)I)
j
=
k
X
j=1
d
j
j
X
`=0
 
j
`
!
Æ
`
e
i`
(A  I)
j `
=
k
X
j=1
d
j
(A  I)
j
+
k
X
j=1
d
j
j 1
X
`=1
 
j
`
!
Æ
`
e
i`
(A  I)
j `
+
0

k
X
j=1
d
j
Æ
j
e
ij
1
A
I
=
k
X
j=1
0

d
j
+
k j
X
`=1
d
j+`
 
j + `
`
!
Æ
`
e
i`
1
A
(A  I)
j
+
0

k
X
j=1
d
j
Æ
j
e
ij
1
A
I:
Suppose d
k
= 
k
and d
j
, j = k   1; :::; 1 are determined by
d
j
= 
j
 
k j
X
`=1
d
j+`
 
j + `
`
!
Æ
`
e
i`
: (9)
Then
k
X
j=1
d
j
(A  (   Æe
i
)I)
j
=
k
X
j=1

j
(A  I)
j
+
0

k
X
j=1
d
j
Æ
j
e
ij
1
A
I; (10)
and the eld of values of this matrix is that of
P
k
j=1

j
(A   I)
j
shifted
by
P
k
j=1
d
j
Æ
j
e
ij
. To express
P
k
j=1
d
j
Æ
j
e
ij
in terms of the 
j
's, note that
if
P
k
j=1

j
(A   I)
j
=
P
k
j=1

j
(A   (   Æe
i
)I   Æe
i
I)
j
is expanded using
the binomial formula, then the expansion involves powers from 1 to k of
A   (   Æe
i
)I plus the term (
P
k
j=1

j
( 1)
j
Æ
j
e
ij
)I. Comparing this with
(10), it follows that
k
X
j=1
d
j
Æ
j
e
ij
=
k
X
j=1

j
( 1)
j+1
Æ
j
e
ij
: (11)
If the origin is on the boundary of F(
P
k
j=1

j
(A   I)
j
), then there is a
line through the origin that separates the eld of values from a half-plane.
5
Suppose 
1
= : : : = 
` 1
= 0 but 
`
6= 0. Choose  so that e
i`
( 1)
`+1

`
lies in the half-plane ontaining F(
P
k
j=1

j
(A   I)
j
) and is orthogonal to
the separating line. Then if the eld of values is shifted in the diretion of
e
i`
( 1)
`+1

`
its losure will exlude the origin. By hoosing Æ > 0 suÆiently
small, one an make the shift term in (11) arbitrarily lose to 
`
( 1)
`+1
Æ
`
e
i`
,
and so one an exlude the origin from the losure of the eld of values of
P
k
j=1
d
j
(A  (  Æe
i
)I)
j
, but this ontradits (8). Therefore eah oeÆient

j
must be 0. 2
Theorem 1 and (5) imply also that
 2 Int(H
k
(A)) if and only if
8
1
; : : : ; 
k
; not all zero, 0 2 Int
0

F
0

k
X
j=1

j
(A  I)
j
1
A
1
A
; (12)
where Int denotes the interior.
If the origin lies outside the losure of the eld of values of an operator
B, then F(B) an be rotated into the right half-plane by multiplying B by
a ertain salar e
i
. Sine the real part of the losure of the eld of values,
ReF(e
i
B) := fRe z : z 2 F(e
i
B)g, is the onvex hull of the spetrum of
the Hermitian part of e
i
B, H(e
i
B) := (e
i
B + e
 i
B

)=2, [6, pp. 9,21℄, it
follows that H(e
i
B) is positive denite. Therefore an equivalent statement
of (4) is
 =2 H
k
(A) if and only if
9 
1
; : : : ; 
k
suh that H(
k
X
j=1

j
(A  I)
j
) is positive denite, (13)
and (5) beomes
 2 H
k
(A) if and only if
8
1
; : : : ; 
k
; H(
k
X
j=1

j
(A  I)
j
) has a nonpositive eigenvalue. (14)
Theorem 1 an be stated as
 2 H
k
(A) if and only if
8 
1
; : : : ; 
k
; 
min
0

H(
k
X
j=1

j
(A  I)
j
)
1
A
 0; and
6
9 ^
1
; : : : ; ^
k
; not all 0, suh that 
min
0

H(
k
X
j=1
^
j
(A  I)
j
)
1
A
= 0; (15)
where 
min
denotes the smallest eigenvalue. In other words, if one hooses

1
; : : : ; 
k
with, say,
P
k
j=1
j
j
j
2
= 1, to maximize 
min
(H(
P
k
j=1

j
(A  I)
j
)),
then this minimum eigenvalue will be positive if  =2 H
k
(A), zero if  2
H
k
(A), and negative if  2 Int(H
k
(A)).
We note one more equivalent denition. In [2℄, Davies introdues the set
Num
k
(A) := fz 2 C : p(z) 2 F(p(A)) 8p 2 P
k
g:
From (5), and the fat that adding a salar times the identity to an operator
shifts its eld of values by that salar, we an write
 2 H
k
(A) if and only if
8 
0
; 
1
; : : : ; 
k
; 
0
2 F
0


0
I +
k
X
j=1

j
(A  I)
j
1
A
: (16)
Sine any polynomial p of degree k or less in A an be written in the form

0
I+
P
k
j=1

j
(A I)
j
for ertain oeÆients 
0
; 
1
; : : : ; 
k
, and then p() = 
0
,
equivalene (16) implies
 2 H
k
(A) () 8 p 2 P
k
; p() 2 F(p(A));
i.e., H
k
(A) = Num
k
(A). It is interesting to note that although H
k
(A) and
Num
k
(A) are the same for bounded operators, only the latter denition
makes sense for unbounded operators.
3 Polynomial Numerial Hulls of Toeplitz Ma-
tries and Operators
Let T denote an innite-dimensional Toeplitz operator:
T =
0
B
B
B
B

a
0
a
 1
a
 2
: : :
a
1
a
0
a
 1
: : :
a
2
a
1
a
0
: : :
.
.
.
.
.
.
.
.
.
.
.
.
1
C
C
C
C
A
: (17)
7
The symbol of T evaluated at a point z is dened as
a(z) := a
0
+
1
X
`=1
a
`
z
`
+
1
X
`=1
a
 `
z
 `
: (18)
We will assume throughout that a(z) 2 L
1
(U), where U := fz : jzj = 1g
denotes the unit irle, so that the matrix in (17) orresponds to a bounded
linear operator on `
2
(see, e.g., [1℄).
The polynomial numerial hull of degree 1 for T , i.e., the losure of the
eld of values, is known [7℄:
H
1
(T ) = o (a(U)) ; (19)
where o() denotes the onvex hull.
Theorem 2. The polynomial numerial hull of degree k  1 for T satises
o (a(U))  H
k
(T )  po
k
(a(U)) : (20)
Here po
k
() denotes the polynomially onvex hull of degree k, dened for
any ompat set S  C as
po
k
(S) := fz 2 C : jp(z)j  max
2S
jp()j 8p 2 P
k
g:
Proof: The left inlusion follows from (19) and the fat that H
k
(T ) 
H
1
(T ). To establish the right inlusion, let p(T ) be any polynomial in T . By
the spetral mapping theorem, (p(T )) = p((T )), where () denotes the
spetrum. By the Hartman-Wintner theorem (see, e.g., [1, Theorem 1.25,
p. 27℄), the essential spetrum of T ontains the range of its symbol. Hene
kp(T )k  sup
2(p(T ))
jj  sup
jzj=1
jp(a(z))j :
The result then follows from denition (1) of H
k
(T ). 2
In words, the polynomial numerial hull of degree k for an innite-dimensional
Toeplitz operator lies somewhere between the polynomially onvex hull of de-
gree k and the ordinary onvex hull of the image, under the symbol, of the
unit irle.
8
Let T
N
be the N by N Toeplitz matrix onsisting of the upper left blok
of the operator in (17):
T
N
=
0
B
B
B
B
B
B
B

a
0
a
 1
a
 2
: : : a
 (N 1)
a
1
a
0
a
 1
: : : a
 (N 2)
a
2
a
1
a
0
: : : a
 (N 3)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
a
N 1
a
N 2
a
N 3
: : : a
0
1
C
C
C
C
C
C
C
A
: (21)
It is known that if T is banded or, more generally, if a(z) is pieewise ontin-
uous on the unit irle, then the polynomial numerial hull of degree 1 (i.e.,
the eld of values) of T
N
approahes that of T as N ! 1 [3, 5, 10℄. More-
over, if p(T
N
) is any polynomial in T
N
and if a(z) is pieewise ontinuous on
the unit irle, then it follows from results in [10, Theorem 2℄ that
p-lim
N!1
F(p(T
N
)) = u-lim
N!1
F(p(T
N
)) = F(p(T )): (22)
Here u-lim
N!1
S
N
, for a sequene of sets S
N
, denotes the set of all limits of se-
quenes of points fs
N
2 S
N
g
1
N=1
, while p-lim
N!1
S
N
denotes all limits of sub-
sequenes fs
N
`
2 S
N
`
g
1
`=1
, where (N
1
; N
2
; : : :) is a subsequene of (1; 2; : : :).
Clearly u-lim
N!1
S
N
 p-lim
N!1
S
N
. Note that if  2 Int(F(p(T ))), then
(22) implies that for N large enough,  2 Int (F(p(T
N
))), sine these sets are
onvex and they ontain sequenes of points approahing points on a irle
about  in Int(F(p(T ))). [See, for example, [11, Prop. 4.15℄.℄
Theorem 3. Assume a(z) in (18) is pieewise ontinuous on the unit irle.
For any xed degree k  1,
H
k
(T )  p-lim
N!1
H
k
(T
N
)  u-lim
N!1
H
k
(T
N
)  l (Int (H
k
(T ))) ; (23)
where l() denotes the losure.
Proof: First suppose 
N
`
2 H
k
(T
N
`
) is a onvergent partial sequene
with 
N
`
!  as ` ! 1. Then it follows from (5) that for all 
1
; : : : ; 
k
,
0 2 F(
P
k
j=1

j
(T
N
`
 
N
`
I)
j
). Sine the eld of values is a ontinuous funtion
of the operator, it follows that for eah 
1
; : : : ; 
k
, there is a sequene of points
s
N
`
2 F(
P
k
j=1

j
(T
N
`
  I)
j
) suh that s
N
`
! 0 as ` ! 1. By (22), this
implies that 0 2 F(
P
k
j=1

j
(T   I)
j
), and hene by (5),  2 H
k
(T ). This
9
establishes the rst inlusion in (23). The seond inlusion is lear from the
denitions of u-lim and p-lim.
Now let  be any point in Int(H
k
(T )). Then we know from (12) that
for all 
1
; : : : ; 
k
with, say,
P
k
j=1
j
j
j
2
= 1, 0 2 Int

F

P
k
j=1

j
(T   I)
j

.
It follows from (22) that there exists n  n(
1
; : : : ; 
k
) suh that for all
N  n, 0 2 Int(F(
P
k
j=1

j
(T
N
  I)
j
)). Sine the oeÆients ome from a
ompat set and sine the eld of values is a ontinuous funtion of 
1
; : : : ; 
k
,
there exist oeÆients ^
1
; : : : ; ^
k
for whih the required minimum n-value,
n^  n(^
1
; : : : ; ^
k
), is maximal and nite. Therefore, for N  n^ and for all

1
; : : : ; 
k
with
P
k
j=1
j
j
j
2
= 1, 0 2 Int(F(
P
k
j=1

j
(T
N
 I)
j
)). Hene by (12),
 2 Int(H
k
(T
N
)). This shows that every interior point of H
k
(T ) eventually
lies in H
k
(T
N
) and hene that every limit of interior points of H
k
(T ) is a
limit of points in H
k
(T
N
). This proves the third inlusion in (23). 2
The preeding theorems leave open the possibility that H
k
(T ) ontains
isolated points or urves between the region enlosed by a(U) and o(a(U)),
and, in this ase, it is not known if there is a sequene or partial sequene
of points in H
k
(T
N
) onverging to these points. In the simplest ase, where
the region enlosed by a(U) is a onvex set, Theorems 2 and 3 imply that
p-lim
N!1
H
k
(T
N
) = u-lim
N!1
H
k
(T
N
) = H
k
(T ) = o(a(U)).
Finally, note that Theorem 3 makes no diret use of Toeplitz properties;
it uses only relation (22) and properties of the polynomial numerial hull.
Hene for any sequene of matries or operators A
N
satisfying
p-lim
N!1
F(p(A
N
)) = u-lim
N!1
F(p(A
N
)) = F(p(A));
for all polynomials p, the analogue of relation (23) will hold, namely,
H
k
(A)  p-lim
N!1
H
k
(A
N
)  u-lim
N!1
H
k
(A
N
)  l (Int (H
k
(A))) ;
for any xed k  1.
Aknowledgment: The authors thank Nik Trefethen for helpful omments
on a draft of this paper.
Referenes
[1℄ A. Botther and B. Silbermann, Introdution to Large Trunated Toeplitz
Matries, Springer, 1999.
10
[2℄ E. B. Davies, Spetral bounds using higher order numerial ranges,
manusript to appear, 2004.
[3℄ M. Eiermann, Fields of values and iterative methods, Lin. Alg. Appl.
180 (1993), pp. 167{197.
[4℄ A. Greenbaum, Generalizations of the eld of values useful in the study
of polynomial funtions of a matrix, Lin. Alg. Appl. 347 (2002), pp. 233{
249.
[5℄ A. Greenbaum, Some theoretial results derived from polynomial numer-
ial hulls of Jordan bloks, ETNA 18 (2004), pp. 81{90.
[6℄ R. A. Horn and C. R. Johnson, Topis in Matrix Analysis, Cambridge
University Press, 1991.
[7℄ E. M. Klein, The numerial range of a Toeplitz operator, Pro. Amer.
Math. So. 35 (1972), pp. 101{103.
[8℄ O. Nevanlinna, Convergene of Iterations for Linear Equations,
Birkhauser, Basel, 1993.
[9℄ O. Nevanlinna, Hessenberg matries in Krylov subspaes and the om-
putation of the spetrum, Numer. Funt. Anal. and Optimiz., 16 (1995),
pp. 443{473.
[10℄ S. Roh, Numerial ranges of large Toeplitz matries, Lin. Alg. Appl.
282 (1998), pp. 185{198.
[11℄ R. T. Rokafellar and J. B. Wets, Variational Analysis, Springer, 1998.
11


Some Equivalent Characterizations of the Polynomial Numerical Hull of Degree $k$

Abstract

Similar works

Full text

Available Versions

Oxford University Research Archive

Mathematical Institute Eprints Archive

Oxford University Research Archive (ORA)

Some Equivalent Characterizations of the Polynomial Numerical Hull of Degree kkk

Abstract

Similar works

Full text

Available Versions

Oxford University Research Archive

Mathematical Institute Eprints Archive

Oxford University Research Archive (ORA)

Some Equivalent Characterizations of the Polynomial Numerical Hull of Degree $k$