On Numerical Ranges of Nilpotent Elements of C*-Algebra

A. Abdollahi, M.T. Heydari and M. Moosavi Department of Mathematics, Shiraz University, Shiraz 71454, Iran

Abstract: Problem statement: Let A be a C*-algebra with unit 1. For each a ∈A, let V(a), ν(a) and

ν0(a) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. Approach: If a is a nilpotent element of A with the power of n nilpotency n, i.e., a = 0, and ν(a) = (n-1) ν0(a). Results: We proved that V(a) = bW(A n), where b is a scalar and A n is the strictly upper triangular n-by-n matrix with all entries above the main diagonal equal to one. Conclusion/Recommendations: We also completely determined the numerical range of such elements, by determining the numerical range of W(A n) and showed that the boundary of it does not contain any arc of circle.

Key words: Numerical range, numerical radius, C*-algebra, nilpotent

INTRODUCTION A complete survey on numerical range can be found in the books by Bonsall and Duncan [2,3] and the Let A be a C*-algebra with unit 1 and let S be the book by Gustafon and Rao [5] and we refer the reader to state space of A, i.e., S = { ϕ∈A*: ϕ≥0, ϕ(1) = 1}. For these books for general information and background. [6] each a ∈A, the C*-algebra numerical range V(a) and In 1992, Haagerup and de la Harpe have proved numerical radius ν(a) is defined, respectively, by: the following sharp estimate for the numerical radius of a nilpotent operator N: V(a): = { ϕ(a): ϕ∈S} and ν(a): = { |z|: z ∈V(a)} π  w(N)≤ N cos   It is well known that V(a) is non empty, compact n+ 1  and convex subset of the complex plane, V( α1+ βa) = α+βV(a) for a∈A and α,β∈ ℂ and if where, n is the power nilpotency of n. In 2004, Karaev z∈V(a), z≤ a [2] . give another proofs of the Haagerup-de la Harpe inequality. The notion of numerical range or the classical field In [4] the researchers have shown that if A is a of values was first introduced by Toeplitz in 1918 for nonzero nilpotent operator with the power of nilpotency matrices. This concept were independently extended by n, with w(A) ≤(n-1)w 0(A) and if A attains its numerical G. Lumer and F. Bauer in sixties to a bounded linear radius then the following conditions are equivalent: operator on an arbitrary Banach space. In 1975, Lightbourne and Martin [9] have extended this concept • w(A) = (n-1)w 0(A) by employing a class of seminorms generated by a • A is unitarily equivalent to an operator of the family of supplementary projections. form ηA ⊕A′, where η is a scalar satisfying As an example, let A be the C*-algebra of all n |η| = 2w 0(A) and A ′ is some other operator: bounded linear operators on a complex Hilbert space H and T ∈A. It is well known that V(T) is the closure of • W(A) = bW(A n) W(T), where: Where: WT( ) := {Tx,x :x ∈ H,x = 1} b = A scalar

0 1⋯ 1  is the usual numerical range of the operator T. In this   0 ⋱ ⋮ special case we denote the numerical radius of T and A : =   n ⋱ 1  the distance from the origin to the boundary of its   0  numerical range by w(T) and w 0(T), respectively. Corresponding Author: A. Abdollahi, Department of Mathematics, Shiraz University, Shiraz 71454, Iran 348 J. Math. & Stat., 5 (4): 348-351, 2009

MATERIALS AND METHODS RESULTS AND DISCUSSION

Let a be a nilpotent with the power of nilpotency Theorem 1: The boundary of W(A n) is: n≥1, i.e., a n = 0 and ν(a) = (n-1) ν (a), where ν (a) 0 0 ∂ γ γ denotes the distance from the origin to the boundary W(A n) = 1+ 2 of its numerical range. In this study we show that Where: V(a) = bW(A n), where b is a scalar and by determining the boundary of numerical range of A n, we show that − π π γ=−+1 3 + 1  the ∂V(a) not inclusive a circle section. Actually, this 2 (t) i cot()t cot()  2 2 n 2n  study is an extension of the Haagerup-de and la Harpe [4] inequality and the research of an earlier study by Gau for 0 ≤t≤1 and: to the C*-algebra numerical range. iθ γ1(θ) = ( λ(θ)+i λ′(θ))e A short survey on W(A n): For the study of numerical ranges of finite matrices, the matrix-theoretic properties n− 1 which λ()0 = and for -π≤θ≤π, θ ≠ 0: can be exploited to yield special tools which are not 2 available for general operators. One important way to ∂ yield W(A) is the Kippenhahn's result that the 1 θ   numerical range of A coincides with the convex hull of λθ=() sin( θ )cot  −θ cos( )  2 n  the real points of the dual curve of det(xReA + ylmA +   [4] zl) = 0 . On the other hand, a parametric representation ∂W(A ) is differentiable at each points. of the boundary W(A) can also be obtained from the n − θ The curvature function of the boundary of largest eigenvalue of Re(e i A) yielding useful numerical range of A n, ∂W(A n), is: information on W(A). × λ θ For any n n matrix A, let ( ) denote the 1 θ θ θ  −iθ R()θ= csc3 −θ sin()cos( ) −θ ncos()sin( ) 2     maximum eigenvalue of Re(e A). It is well known nn n n  that λ(θ) is an analytic function of θ (possibly except ℂn for some isolated points) and a unit vector in is Proof: First we want to compute the λ(θ). Put M n = iθ such that belong to ∂W(A) ∩Lθ if and only if Re(e An)-λIn and P n(λ) = detM n. Therefore: − θ Re(e i A)x = λ(θ)x [4] . Also ∂W(A) admits a parametric −θi −θ i −θ i  representation: −λ e e⋯ e   2 2 2  x( θ) = λ(θ) cos( θ)-λ′(θ) sin ( θ) eiθ e −θ i e −θ i  −λ ⋯  2 2 2 y(θ) = λ(θ) sin ( θ)+ λ′(θ) cos ( θ)   P (λ ) = det iθθ i −θ i n e e e  −λ ⋯  (again, with possible exception of finitely many points). 2 2 2  The curvature and radius of curvature of ∂W(A) at p = ⋮ ⋮ ⋮⋱⋮  θ θ iθ i θ i θ  (x( ), y( )) are equals: e e e ⋯ −λ  2 2 2  1 K(θ ) = λθ( ) +λ ''( θ ) by adding the -1 multiple of any row to before row we have: and  iθ − i θ  −λ−e λ+ e ⋯ R( θ) = λ(θ)+ λ′′(θ),  0 0   2 2  iθ − i θ  e e  respectively.  0−λ− λ+ ⋯ 0  As we mentioned, if A is a nilpotent operator on a  2 2  λ = −i θ P(n ) det  e  Hilbert space H with nilpotency n that attaint its  0 0−λ − ⋯ 0  2 numerical radius and w(A) = (n-1)w 0(A) then    ⋮ ⋮ ⋮⋱⋮  W(A) = bW(A n), for some b. Hence for determining  iθ i θ i θ  W(A) it is enough to compute W(A n). The following  e e e ⋯ −λ  theorem can be help us to find W(A).  2 2 2  349 J. Math. & Stat., 5 (4): 348-351, 2009

Important, but easy to check, is the fact that when By direct calculation we have: expanding this determinant about the first column, we have the following Recursion formula: θ θ λθ=1 θ − 1 θ2 +θ  '( ) cos( )cot( ) sin( )csc ( ) sin( )  2 nn n  iθ  i θ−θ i λ=−λ−e λ+−n1− e λ+ e n1 − ≥ P()n  P()(1) n− 1 ( ),n1 2  2 2 for -π≤θ≤π, θ ≠ 0 and λ′ (0) = 0 also:

λ with initial condition P o( ) = 1. θ2 θ  −θsin( )cot( ) − cos( θ )csc2 ( ) By solving this recursion formula we have: 1   λ''( θ ) = n n n  θ θ 2 1 3  n1+ i θ −θ i + θ θ −   2 sin( )cos( )csc ( )cos( ) λ=( 1)−i θ λ+ e n −λ+ i θ e n n n  P()n iθ − i θ  e( )e( )n  ee−  2 2  for -π≤θ≤π, θ ≠ 0 and λ′′(0) = 0 So the curvature θ ≠ π ∈ ℤ if m , for each m and: function of the boundary of numerical range of A n, ∂W(A ), is: n+ 1 n (− 1) 1 − λ=() −λ− λ+ n 1 P()n n21( ) 2 2 1 θ θ θ R(θ= ) csc3 ( )[sin( θ )cos( ) −θ n cos( )sin( )] nn2 n n θ π ∈ ℤ whenever = m , for some m . λ By solving P n( ) = 0, in the first case we have: Now the proof is completed. This proof will help in discussing the following n eiθ  corollary: λ +  2 = 2i θ − θ  e e i  Corollary 1: By radius of curvature, ∂W(A ) does not λ +  n 2  contain any arc of circle.

Which implies: C*-algebra numerical range of nilpotent elements: As we mentioned in the introduction, let A be a C*- π + θ λ=1 θ k −θ=  − algebra with unit 1 and let S be the state space of A, k sin cot( ) cos  ,k 0,1,...,n 1 2 n  i.e., S = { ϕ∈A*: ϕ≥0, ϕ (1) = 1}. For each a ∈A, the C*- algebra numerical range is defined by: By a simple computation we see that for θ ≠ 0, λ(θ) = λ0≥λk, for each k and so λ(θ) satisfies (a). This V(a): = { ϕ(a): ϕ∈S} relation with Eq. 1 and inequality sink θ≤ksin θ implies n− 1 1 Analogy we define ν(a) and ν (a) for an element that w(A ) = and w (A ) = , independent from n. 0 n 2 0 n 2 a∈A by: This facts will help in discussing and proving many π θ π of the result below. Therefore let - < < and put: ν=(a) sup{z:z ∈∂ V(a)}

γ θ λ θ λ′ θ iθ 1( ) = ( ( )+i ( ))e and

Also define: ν=(a) dist(0, ∂ V(a)) = inf{z :z ∈∂ V(a)} 0 1− 3 π 1 π γ=−+(t) i( cot( )t + cot( )) 2 2 2 n 2 n Now we turn our attention to nilpotency.

∈ for 0 ≤t≤1. So we have: Theorem 2: Let A be a C*-algebra with unit and a A be a nilpotent element with nilpotency n. Then: ∂W(A ) = γ +γ n 1 2 π ν(a) ≤ a cos( ) n+ 1 It is easy to show that ∂W(A n) is differentiable. 350 J. Math. & Stat., 5 (4): 348-351, 2009

If ν(a) = (n-1) ν0(a), then: REFERENCES

V(a) = bW(A n) 1. Berberian, S.K. and G.H. Orland, 1967. On the closure of the numerical range of an operator. Proc. where, b is a scalar. Am. Math. Soc., 18: 499-503. http://www.jstor.org/stable/2035486 n− 1 π 2. Bonsall, F.F., J. Duncan and N.J. Hitchin, 1971. a≥ b sec 2 n+ 1 Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge ∂V(a) does not contain any arc of circle. University Press, Cambridge, ISBN: 13: 9780521079884, pp: 148. Proof: Let ρ be a state of A. Then there exists a cyclic 3. Bonsall, F.F. and J. Duncan, 1973. Numerical Ranges II. Cambridge University Press, representation πρ of A on a Hilbert space H ρ and a unit Cambridge, ISBN: 10: 052120227, pp: 188. cyclic vector x ρ for H ρ such that: 4. Gau, H.L. and P.Y. Wu, 2008. Numerical ranges of

nilpotent operators. Linear Algebra Applied, ρ 〈πρ ρ ρ〉 ∈ (a) = (a)x , x , a A 429: 716-726. 5. Gustafon, K.E. and K.M. Rao, 1996. The By Gelfand-Naimark theorem the direct sum Numerical Range: The Field of Values of Linear π:a → ⊕π (a) is a faithful representation of A on ∑ρ∈ s ρ Operators and Matrices. 1st Edn., Springer, New the Hilbert space H= ⊕ H [8] . Therefore for each York, ISBN: 10: 038794835X, pp: 189. ∑ρ∈ s ρ ρ∈ ρ ∈ π ⊂ π 6. Haagerup, U. and P. de la Harpe, 1992. The S, (a) W( ρ)(a) W( (a)) and hence V(a) numerical radius of a nilpotent operator on a π contained in W( (a)). On the other hand if x is a unit Hilbert space. Proc. Am. Math. Soc., 115: 371-379. ρ π ∈ vector of H, then the formula (c) = 〈 (c)x, x 〉,c A, http://www.jstor.org/stable/2159255 defines a state on A and hence ρ(a) = 〈π(a)x, x 〉∈V(a) 7. Halmos, P.R., 1982. A Hilbert Space Problem and it follows that: Book. 2nd Edn., Springer, New York, ISBN: 3- 540-90685-1. W(T a) = V(a) 8. Kadison, R.V. and J.R. Ringrose, 1983. Fundamentals of the Theory of Operator Algebras: π [1] where, T a = (a) . Elementary Theory. Academic Press, New York, n n = Also a = 0 implies that Ta 0,T a is a nilpotent operator ISBN: 10: 0123933013, pp: 416. with nilpotency n. Then (i) follows from (1) and (2). 9. Lightbourne, J.H. and R.H. Martin, 1975. Part (iv) follows from Theorem 1. Also w(T a) = (n-1) Projection seminorms and the field of values of w0(T a) and so V(a) = W(T a) = bW(A n), where b is a linear operators. Num. Math., 24: 151-161. DOI: scalar, which implies (ii). Finally (iii) follows from (i) 10.1007/BF01400964 n− 1 and the facts that ν(a) = bw(A ) and w(A ) = . n n 2

CONCLUSION

If A is a C*-algebra with unit and a∈A is a nilpotent element with nilpotency n and ν(a) = (n-

1) ν0(a), then:

V(a) = bW(A n)

where, b is a scalar. Also ∂V(a) does not contain any arc of circle. So, we can completely determine the numerical range of such elements.

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