The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix
Benjamín A. Itzá-Ortiz, Rubén A. Martínez-Avendaño, Hiroshi Nakazato
aa r X i v : . [ m a t h . F A ] F e b THE NUMERICAL RANGE OF A PERIODIC TRIDIAGONAL OPERATORREDUCES TO THE NUMERICAL RANGE OF A FINITE MATRIX
BENJAMÍN A. ITZÁ-ORTIZ, RUBÉN A. MARTÍNEZ-AVENDAÑO, AND HIROSHI NAKAZATO
Dedicated to the memory of Rudolf Kippenhahn (1926 – 2020)
Abstract.
In this paper we show that the closure of the numerical range of a n +1-periodictridiagonal operator is equal to the numerical range of a 2( n + 1) × n + 1) complex matrix. Introduction
Consider A to be a finite set of complex numbers and let a = ( a i ) i ∈ Z be a biinfinitesequence in the total shift space A Z . In [12], the tridiagonal operator A a : ℓ ( Z ) → ℓ ( Z )associated to a is defined as(1) A a = . . . . . .. . . a − a − a a . . .. . . . . . where the square marks the matrix entry at (0 , A = {− , } , the corresponding operator A a is related to the so called “hopping sign model”introduced in [7] and subsequently studied in many other works, such as [1, 2, 3, 4, 5, 6, 8,9, 12], just to name a few. On the other hand, when the alphabet is A = { , } some resultsfor computing the numerical range of A a are presented in [12, 13]. In particular, workin [13] addresses the case when a is an n + 1-periodic sequence. Relying on the fact thatthe closure of the numerical range of A a may be written as the closure of the convex hullof an uncountable union of numerical ranges of certain matrices, in [13] the closure ofthe numerical range of the 2-periodic case is computed by substituting such uncountableunion of numerical ranges by the convex hull of the union of the numerical ranges of justtwo 2 × A a when a is an n + 1 periodic biinfinite sequence.Instead of working with the operators A a , we work with the more general tridiagonaloperators T = T ( a, b, c ) defined in Section 2, since, as can be seen in [13], the computationof the closure of the numerical range of A a is a particular case of that of T . Using a resultof Plaumann and Vinzant [18], we show that the closure of the numerical range of the Date : January 2021.The second author’s research is partially supported by the Asociación Mexicana de Cultura A.C.. n + 1 periodic tridiagonal operator T is the numerical range of a 2( n + 1) × n + 1) matrix(cf. Theorem 2.6).We divide this work in two sections. In Section 1 we briefly introduce the notationand terminologies needed in the rest of the paper. In Section 2 we develop the requiredmachinery, first by computing the Kippenhahn polynomial of the symbol of n + 1 periodctridiagonal operatos T on ℓ ( N ) and then by combining our computations with resultsof Plaumann and Vizant. We will conclude that the closure of the numerical range of T is equal to the numerical range of a 2( n + 1) × n + 1) matrix.1. Preliminaries
In this section we introduce the notation required which will be needed in the followingsections. As usual, the symbols N , N , Z , R and C will denote the set of positive integers,the sets of nonnegative integers, the set of integers, the set of real numbers and the set ofcomplex numbers, respectively.For a given n ∈ N , let a , b and c be ( n + 1)-periodic infinite sequences in A N . We willdenote by T = T ( a, b, c ) the ( n + 1)-periodic tridiagonal operator on ℓ ( N ) given by T = b c a b c a b c . . . . . . . . .a n b n c n a b c . . . . . . . . .a n − b n − c n − a n b n c n . . . . . . . . . . We should observe that T is a bounded operator since the sum of the moduli of the en-tries in each column (and in each row) is uniformly bounded (see, e.g., [15, Example 2.3]).The biinfinite matrix A a is also a bounded operator, as long as the biinfinite sequence a arises from a finite alphabet.If n >
1, for each φ ∈ [0 , π ), following [1, 13] we define the symbol of T , as the follow-ing ( n + 1) × ( n + 1) matrix(2) T φ = b c a e − iφ a b c a b c . . . . . . . . . . . . . . . a n − b n − c n −
00 0 a n − b n − c n − c n e iφ a n b n ;while the symbol of T for n = 1 is the 2 × T φ = b c + a e − iφ a + c e iφ b ! . UMERICAL RANGE OF A TRIDIAGONAL OPERATOR 3
Recall that given a Hilbert space H and a bounded operator A on it, the numerical rangeis defined as the set W ( A ) = {h Ax, x i : k x k = 1 } . The Toeplitz-Haussdorf Theorem establishes that W ( A ) is a bounded convex subset of C (closed, if the Hilbert space is finite dimensional) and hence the closure of the numericalrange can be seen as the intersection of the closed half-spaces containing the numericalrange.Kippenhahn [16] (see also [17]) characterized two vertical support lines of W ( A ) fora given n × n matrix as Re( z ) = λ ( A ) and Re( z ) = λ n ( A ), where λ ( A ) and λ n ( A ) arethe respective largest and least eigenvalues of Re( A ) (recall that Re( A ) := ( A + A ∗ ) andIm( A ) := i ( A − A ∗ )). In fact, if α ∈ W ( A ) then λ n ( A ) ≤ Re( α ) ≤ λ ( A ) (and the equali-ties hold for some points α , α ∈ W ( A )). Since e iθ W ( A ) = W ( e iθ A ) for each θ ∈ [0 , π ),it follows that if α ∈ W ( A ), then e − iθ α ∈ W ( e − iθ A ) and hence Re( e − iθ α ) ≤ λ ( e − iθ A ). Itfollows that the lines Re( e − iθ z ) = λ ( e − iθ A ) are support lines of W ( A ). Hence the con-vex set W ( A ) is uniquely determined by the numbers λ ( e − iθ A ), as θ varies on the inter-val [0 , π ); i.e. W ( A ) is determined by the largest eigenvalue of Re( e − iθ A ), which equalscos( θ )Re( A ) + sin( θ )Im( A ). Thus the numerical range is determined by the largest rootsof the family of characteristic polynomialsdet( tI n − cos( θ )Re( A ) − sin( θ )Im( A )) . The homogeneous polynomial F A ( t, x, y ) = det( tI n + x Re( A ) + y Im( A )) is called the Kip-penhahn polynomial of the matrix A . It clearly follows that two matrices have the samenumerical range if their Kippenhahn polynomials coincide. Furthermore,max { t ∈ R : F A ( t, − cos( θ ) , − sin( θ )) = 0 } = max { Re( e − iθ z ) : z ∈ W ( A ) } for each θ ∈ [0 , π ).2. The Kippenhahn polynomial of the symbol T φ In this section, after some preliminary work, we show that the closure of the numericalrange of a n +1-periodic tridiagonal operator T is the numerical range of a 2( n +1) × n +1)matrix.We will need the following lemma. Lemma 2.1.
Consider the ( n + 1) × ( n + 1) “almost tridiagonal” matrix Λ = λ , λ , . . . λ ,n +1 λ , λ , λ , . . . λ , λ , λ , . . . λ , λ , . . . ... ... ... ... . . . ... ... ... . . . λ n − ,n − λ n − ,n
00 0 0 0 . . . λ n,n − λ n,n λ n,n +1 λ n +1 , . . . λ n +1 ,n λ n +1 ,n +1 , BENJAMÍN A. ITZÁ-ORTIZ, RUBÉN A. MARTÍNEZ-AVENDAÑO, AND HIROSHI NAKAZATO where every λ i,j ∈ C . Then, det( Λ ) equalsdet λ , λ , . . . λ , λ , λ , . . . λ , λ , . . . ... ... ... . . . ... ... . . . λ n − ,n
00 0 0 . . . λ n,n λ n,n +1 . . . λ n +1 ,n λ n +1 ,n +1 − λ ,n +1 λ n +1 , det λ , λ , . . . λ , λ , . . . ... ... . . . ... ... . . . λ n − ,n − λ n − ,n . . . λ n,n − λ n,n + ( − n λ n +1 , λ , λ , · · · λ n − ,n λ n,n +1 + ( − n λ ,n +1 λ , λ , · · · λ n,n − λ n +1 ,n . Proof.
This follows by a long (but straightforward) application of the multilinearity of thedeterminant function and the Laplace Expansion Theorem. (cid:3)
Let us set the following notation for the rest of this paper. For 0 ≤ j < n we define α j = c j + a j +1 , γ j = c j − a j +1 i and α n = a + c n , γ n = a − c n i . We now find an expression for the Kippenhahn polynomial F T φ of the symbol matrix T φ of an arbitrary n + 1-periodic tridiagonal matrix T acting on ℓ ( N ), involving the de-terminants of some tridiagonal matrices. This expression will be useful in what follows. Proposition 2.2.
Let n ∈ N . Consider the symbol T φ , that is, the ( n + 1) × ( n + 1) matrixdefined as in (2) for n ≥ n = 1. Then the Kippenhahn polynomial of T φ is equal to F T φ ( t, x, y ) = G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) + 2( − n Re ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) cos φ − − n Im ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) sin φ, where G n ( t, x, y ) is the determinant of the tridiagonal ( n + 1) × ( n + 1) matrix λ , λ , . . . λ , λ , λ , · · · λ , λ , λ , · · · λ , λ , · · · ... ... ... ... ... ... · · · λ n,n λ n,n +1 · · · λ n +1 ,n λ n +1 ,n +1 , UMERICAL RANGE OF A TRIDIAGONAL OPERATOR 5 and, where we set H n ( t, x, y ) = 1 when n = 1, and, for n ≥
2, we set H n ( t, x, y ) to be thedeterminant of ( n − × ( n −
1) tridiagonal matrix λ , λ , · · · λ , λ , λ , · · · λ , λ , · · · ... ... ... ... ... · · · λ n − ,n − λ n − ,n · · · λ n,n − λ n,n . Here we have set, for 1 ≤ j ≤ n + 1, λ j,j = t + Re( b j − ) x + Im( b j − ) y, and for 1 ≤ j ≤ n , λ j,j +1 = α j − x + γ j − i y and λ j +1 ,j = α j − x + γ j − y. Proof.
We divide the proof in two cases. For n +1 = 2, by computing the real and imaginaryparts of the matrix T φ in (3), we obtain that the 2 × tI + x Re( T φ )+ y Im( T φ ) is givenby t + Re( b ) x + Im( b ) y α x + γ y + ( α x + γ y ) e − iφ ( α x + γ y ) + ( α x + γ y ) e iφ t + Re( b ) x + Im( b ) y ! , where α , α , γ and γ are as defined above. The determinant of this matrix can besimplified to F T φ ( t, x, y ) = ( t + Re( b ) x + Im( b ) y )( t + Re( b ) x + Im( b ) y ) − | α x + γ y | − | α x + γ y | − (cid:16) ( α x + γ y )( α x + γ y ) e iφ (cid:17) = ( t + Re( b ) x + Im( b ) y )( t + Re( b ) x + Im( b ) y ) − | α x + γ y | − | α x + γ y | − α x + γ y )( α x + γ y )) cos φ + 2Im (( α x + γ y )( α x + γ y )) sin φ = G ( t, x, y ) − | α x + γ t | H ( t, x, y ) − α x + γ y )( α x + γ y )) cos φ + 2Im (( α x + γ y )( α x + γ y )) sin φ, as desired.Now, for the case n + 1 ≥
3, by computing the real and imaginary parts of the matrix T φ in (2), we can observe that tI n +1 + x Re( T φ ) + y Im( T φ ) is the matrix λ , λ , . . . λ ,n +1 λ , λ , λ , · · · λ , λ , λ , · · · λ , λ , · · · ... ... ... ... ... ... · · · λ n,n λ n,n +1 λ n +1 , · · · λ n +1 ,n λ n +1 ,n +1 , where we have now set λ ,n +1 = ( α n x + γ n y ) e − iφ and λ n +1 , = ( α n x + γ n y ) e − iφ . The above matrix is tridiagonal, except for the upper-right and bottom-left corners.
BENJAMÍN A. ITZÁ-ORTIZ, RUBÉN A. MARTÍNEZ-AVENDAÑO, AND HIROSHI NAKAZATO
We can compute the determinant of the matrix polynomial tI n +1 + x Re( T φ ) + y Im( T φ ) byusing Lemma 2.1 obtaining F T φ ( t, x, y ) = det( tI n +1 + x Re( T φ ) + y Im( T φ ))= G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y )+ ( − n ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) e iφ + ( − n ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) e − iφ = G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) + 2( − n Re ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) e iφ . Computing the real part of the last term above, we obtain the equation F T φ ( t, x, y ) = G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) + 2( − n Re ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) cos φ − − n Im ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) sin φ, which completes the proof. (cid:3) For every n ∈ N and for a fixed point ( x, y ) ∈ R , the angle φ ∈ [0 , π ) is involved onlyin the constant term (with respect to the variable t ) of the polynomial F T φ ( t, x, y ). Fur-thermore, for every ( x, y ) ∈ R and for every φ ∈ [0 , π ), the polynomial F T φ ( t, x, y ), seenas a polynomial in t , has n + 1 real roots, counting multiplicities, as it is the characteristicpolynomial of the Hermitian matrix − x Re( T φ ) − y Im( T φ ). The following lemma will beuseful later when applied to the polynomial F T φ . Lemma 2.3.
Let F ( t : φ ) be a family of polynomials in R [ t ] given by the expression F ( t : φ ) = t n +1 + p n t n + . . . + p t + p − u cos φ − v sin φ, where φ ∈ [0 , π ). Assume that the polynomial F ( t : φ ) has n + 1 real roots countingmultiplicities for any angle φ ∈ [0 , π ). Let φ , φ ∈ [0 , π ) be such that u cos φ + v sin φ = −√ u + v and u cos φ + v sin φ = √ u + v . Then max { max { t ∈ R : F ( t : φ ) = 0 } : 0 ≤ φ < π } = max { t ∈ R : F ( t : φ ) = 0 } , and min { max { t ∈ R : F ( t : φ ) = 0 } : 0 ≤ φ < π } = max { t ∈ R : F ( t : φ ) = 0 } . Proof.
Define p ( t ) as p ( t ) = t n +1 + p n t n + . . . + p t + p . Observe that, by assumption, the equation p ( t ) = u cos φ + v sin φ UMERICAL RANGE OF A TRIDIAGONAL OPERATOR 7 has n +1 real solutions (counting multiplicities) for every φ ∈ [0 , π ). For some φ ∈ [0 , π ),we have u cos φ + v sin φ = 0, and hence p has n + 1 real roots (counting multiplicities) andthe derivative of p has n real roots (counting multiplicities). Let r be the largest root of p ′ ( t ). Hence, p is increasing on the interval [ r , ∞ ) and the equations p ( t ) = u cos φ + v sin φ have a unique solution on the interval [ r , ∞ ).Observe that for every φ ∈ [0 , π ) −√ u + v ≤ u cos φ + v sin φ ≤ √ u + v ;equality occurs on the left-hand-side inequality at φ while equality occurs on the right-hand-side inequality at φ .For each φ ∈ [0 , π ), consider the numbermax { t ∈ R : p ( t ) = u cos φ + v sin φ } . Since the function p is increasing on [ r , ∞ ), the largest of these numbers, when φ varies,occurs when t is the largest solution of the equation p ( t ) = √ u + v . Hence we havemax { max { t ∈ R : F ( t, φ ) = 0 } : 0 ≤ φ < π } = max { t ∈ R : F ( t, φ ) = 0 } . Analogously, the smallest, when φ varies in [0 , π ), among the largest solutions t of theequations p ( t ) = u cos φ + v sin φ, occurs when t is the largest solution of the equation p ( t ) = −√ u + v . Hence we havemin { max { t ∈ R : F ( t, φ ) = 0 } : 0 ≤ φ < π } = max { t ∈ R : F ( t, φ ) = 0 } . (cid:3) The previous lemma plays a central role in the proof of the following result.
Proposition 2.4.
Let n ∈ N . Suppose that T ( a, b, c ) is an n + 1-periodic tridiagonal op-erator acting on ℓ ( N ). Let G n and H n be as in Proposition 2.2 and let P be the realhomogeneous polynomial of degree 2( n + 1) given by P ( t, x, y ) = (cid:16) G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) (cid:17) − n Y j =0 (cid:12)(cid:12)(cid:12) α j x + γ j y (cid:12)(cid:12)(cid:12) . Then P ( t, ,
0) = t n +1) andsup n Re( e − iθ z ) : z ∈ W ( T ( a, b, c )) o = max { t ∈ R : P ( t, − cos θ, − sin θ ) = 0 } , for each θ ∈ [0 , π ). BENJAMÍN A. ITZÁ-ORTIZ, RUBÉN A. MARTÍNEZ-AVENDAÑO, AND HIROSHI NAKAZATO
Proof.
It is trivial to check that P ( t, ,
0) = t n +1) . Now, let F ( t : φ ) = F T φ ( t, x, y ), where weknow by Proposition 2.2 that F T φ ( t, x, y ) = G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) − u cos φ − v sin φ, where u = − − n Re ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) and v = 2( − n Im ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) . Notice that u + v = 4Re ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) + 4Im ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) = 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( α n x + γ n y ) n − Y j =0 ( α j x + γ j y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =4 n Y j =0 (cid:12)(cid:12)(cid:12) α j x + γ j y (cid:12)(cid:12)(cid:12) . The polynomial F ( t : φ ) has the form outlined in Lemma 2.3 and, as was mentionedbefore Lemma 2.3, it has n + 1 real roots, counting multiplicities. Hence, by Lemma 2.3,for φ and φ satisfying u cos( φ ) + v sin( φ ) = −√ u + v , u cos( φ ) + v sin( φ ) = √ u + v , we have that max { max { t : F ( t : φ ) = 0 } : 0 ≤ φ < π } = max { t : F ( t : φ ) = 0 } , and min { max { t : F ( t : φ ) = 0 } : 0 ≤ φ < π } = max { t : F ( t : φ ) = 0 } . Notice that F ( t : φ ) · F ( t : φ ) = (cid:16) G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) − ( u cos( φ ) + v sin( φ )) (cid:17) · (cid:16) G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) − ( u cos( φ ) + v sin( φ )) (cid:17) = (cid:16) G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) (cid:17) − (cid:16) √ u + v (cid:17) = (cid:16) G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) (cid:17) − ( u + v )= P ( t, x, y ) . UMERICAL RANGE OF A TRIDIAGONAL OPERATOR 9
We also have, for each θ ∈ [0 , π ), thatmax { t ∈ R : F T φ ( t, − cos θ, − sin θ ) = 0 , ≤ φ < π } = max n max n t ∈ R : F T φ ( t, − cos θ, − sin θ ) = 0 o : 0 ≤ φ < π o = max n t ∈ R : F T φ ( t, − cos θ, − sin θ ) = 0 o = max { t ∈ R : P ( t, − cos θ, − sin θ ) = 0 } . (4)The last equality follows since the roots of P ( t, − cos θ, − sin θ ) are those of F ( t : φ ) = F T φ ( t, − cos θ, − sin θ ) and F ( t : φ ) = F T φ ( t, − cos θ, − sin θ ), so by the choice of φ and φ ,the largest root of P ( t, − cos θ, − sin θ ) is the largest root of F T φ ( t, − cos θ, − sin θ ).By the definition of the Kippenhahn polynomial, we havemax n t ∈ R : F T φ ( t, − cos( θ ) , − sin( θ )) = 0 o = max n Re( e − iθ z ) : z ∈ W ( T φ ) o . and hence we obtainmax n t ∈ R : F T φ ( t, − cos( θ ) , − sin( θ )) = 0 , ≤ φ < π o = max n Re( e − iθ z ) : z ∈ W ( T φ ) , ≤ φ < π o . (5)Lastly, the equality(6) sup n Re( e − iθ z ) : z ∈ W ( T ( a, b, c )) o = max n Re( e − iθ z ) : z ∈ W (cid:16) T φ (cid:17) , ≤ φ < π o follows from Theorem 2.8 in [13]. Putting together equations (4), (5) and (6), we obtainthe desired conclusion. (cid:3) The following definition will be useful.
Definition 2.5.
Suppose that Q ( t, x, y ) is a real homogeneous polynomial in 3 variables t, x, y of degree m with Q (1 , , >
0. If the equation Q ( t, x , y ) = 0 in t has m real solutionscounting multiplicities for any ( x , y ) ∈ R with x + y >
0, we say that Q is hyperbolic (with respect to (1 , , Q ( t, − cos θ, − sin θ ) = 0in t has m real solutions for any angle 0 ≤ θ < π ”. Theorem 2.6 (Plaumann and Vinzant [18]) . Suppose that Q ( t, x, y ) is a real homogeneoushyperbolic polynomial of degree m with Q (1 , ,
0) = 1. Then there exists an m × m complexmatrix A satisfying Q ( t, x, y ) = det( tI m + x Re( A ) + y Im( A )) . Remark.
Helton and Vinnikov [11] (cf. [10]) proved a result stronger than the abovetheorem which guarantees that we can construct an m × m complex symmetric matrix A satisfying a similar property. In this paper we do not use the symmetry of the matrix A .Depending on the above Theorem 2.6, we obtain the main theorem of this paper. Theorem 2.7.
Suppose that T ( a, b, c ) is an n + 1-periodic tridiagonal operator acting on ℓ ( N ). Then there exists a 2( n + 1) × n + 1) complex matrix A such that W ( T ( a, b, c )) = W ( A ) where the matrix A is chosen so that it satisfies F A ( t, x, y ) = (cid:16) G n ( t, x, y ) − | α n x + γ n y | H n ( t, x, y ) (cid:17) − n Y j =0 (cid:12)(cid:12)(cid:12) α j x + γ j y (cid:12)(cid:12)(cid:12) , where the polynomials G n and H n are as in Proposition 2.2. Proof.
By Theorem 2.6, there exists a 2( n + 1) × n + 1) matrix A such that P ( t, x, y ) = F A ( t, x, y ), where P is the homogeneous polynomial in Proposition 2.4. But also, by thesame proposition,sup n Re( e − iθ z ) : z ∈ W ( T ( a, b, c )) o = max { t ∈ R : F A ( t, − cos θ, − sin θ ) = 0 } = max n Re( e − iθ z ) : z ∈ W ( A ) o for each θ ∈ [0 , π ), and hence the closure of the numerical range of T ( a, b, c ) equals thenumerical range of A . (cid:3) In some cases, the matrix A can be found explicitly, as the next example shows. Example 2.8.
Let a be the 2-periodic sequence with period word 01, let b be the constant0 sequence and let c be the constant 1 sequence. If A = − − then W ( T ( a, b, c )) = W ( A ) Proof.
It is a straightforward computation that the polynomial P in Proposition 2.4 equals P ( t, x, y ) = (cid:18) t − x − (cid:16) x + y (cid:17)(cid:19) − x ( x + y ) . But a computation also shows that F A ( t, x, y ) = P ( t, x, y ) and hence, by Theorem 2.7, wehave W ( T ( a, b, c )) = W ( A ). (cid:3) Observe that, in this case, this shows that the closure of the numerical range of T ( a, b, c )equals the convex hull of the numerical ranges of the matrices ! and − − ! , recovering Theorem 3.6 in [13]. In the paper [14] we explore further conditions underwhich the matrix A can be explicitly found. References [1] N. Bebiano, J. da Providência, and A. Nata. The numerical range of banded periodic Toeplitz operators.
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