REVERSE INEQUALITIES FOR THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.

1. Introduction

Let (H; ⟨·, ·⟩) be a complex Hilbert space. The numerical range of an operator T is the subset of the complex numbers ℂ given by [1, p. 1]:

W (T) = {⟨T x, x⟩ , x ∈ H, ∥x∥ = 1} .

The following properties of W (T) are immediate:

(i) W (αI + βT) = α + βW (T) for α, β ∈ ℂ; (ii) W (T^{∗}) = {λ̄, λ ∈ W (T)} , where T^{∗} is the adjoint operator of T; (iii) W (U^{∗}T U) = W (T) for any unitary operator U. The following classical fact about the geometry of the numerical range [1, p. 4]

may be stated:

Theorem 1 (Toeplitz-Hausdorff). The numerical range of an operator is convex.

An important use of W (T) is to bound the spectrum σ (T) of the operator T [1, p. 6]:

Theorem 2 (Spectral inclusion). The spectrum of an operator is contained in the closure of its numerical range.

The self-adjoint operators have their spectra bounded sharply by the numerical range [1, p. 7]:

Theorem 3. The following statements hold true:

(i) T is self-adjoint iff W (T) is real;

(ii) If T is self-adjoint and W (T) = [m, M] (the closed interval of real numbers

m, M), then ∥T ∥ = max {|m| , |M|} . (iii) If W (T) = [m, M] , then m, M ∈ σ (T) .

The numerical radius w (T) of an operator T on H is given by [1, p. 8]:

(1.1)  w (T) = sup {|λ| , λ ∈ W (T)} = sup {|⟨T x, x⟩| , ∥x∥ = 1} .

Date: 29 August, 2005.

2000 Mathematics Subject Classification. 47A12.

Key words and phrases. Numerical range, Numerical radius, Bounded linear operators, Hilbert spaces.

Obviously, by (1.1), for any x ∈ H one has

(1.2)  |⟨T x, x⟩| ≤ w (T) ∥x∥^{2} .

It is well known that w (·) is a norm on the Banach algebra B (H) of all bounded linear operators T : H → H, i.e.,

(i) w (T) ≥ 0 for any T ∈ B (H) and w (T) = 0 if and only if T = 0; (ii) w (λT) = |λ| w (T) for any λ ∈ ℂ and T ∈ B (H) ; (iii) w (T + V) ≤ w (T) + w (V) for any T, V ∈ B (H) .

This norm is equivalent with the operator norm. In fact, the following more precise result holds [1, p. 9]:

Theorem 4 (Equivalent norm). For any T ∈ B (H) one has

(1.3)  w (T) ≤ ∥T ∥ ≤ 2w (T) .

Let us now look at two extreme cases of the inequality (1.3). In the following r (t) := sup {|λ| , λ ∈ σ (T)} will denote the spectral radius of T and σ_{p} (T) = {λ ∈ σ (T) , T f = λf for some f ∈ H} the point spectrum of T.

The following results hold [1, p.10]:

Theorem 5. We have

(i) If w (T) = ∥T ∥ , then r (T) = ∥T ∥ .

(ii) If λ ∈ W (T) and |λ| = ∥T ∥ , then λ ∈ σ_{p} (T) .

To address the other extreme case w (T) = \frac{1}{2} ∥T ∥ , we can state the following sufficient condition in terms of (see [1, p. 11])

R (T) := {T f, f ∈ H} and R (T^{∗}) := {T^{∗}f, f ∈ H} .

Theorem 6. If R (T) ⊥ R (T^{∗}) , then w (T) = \frac{1}{2} ∥T ∥ .

It is well-known that the two-dimensional shift

S_{2} = [ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ] ,

has the property that w (T) = \frac{1}{2} ∥T ∥ .

The following theorem shows that some operators T with w (T) = \frac{1}{2} ∥T ∥ have S_{2} as a component [1, p. 11]:

Theorem 7. If w (T) = \frac{1}{2} ∥T ∥ and T attains its norm, then T has a two-dimensional reducing subspace on which it is the shift S_{2}.

For other results on numerical radius, see [2], Chapter 11.

The main aim of the present paper is to point out some upper bounds for the nonnegative difference

∥T ∥ − w (T) (∥T ∥^{2} − (W (T))^{2})

under appropriate assumptions for the bounded linear operator T : H → H.

2. The Results

The following results may be stated:

Theorem 8. Let T : H → H be a bounded linear operator on the complex Hilbert space H. If λ ∈ ℂ\ {0} and r > 0 are such that

(2.1)  ∥T − λI∥ ≤ r,

where I : H → H is the identity operator on H, then

(2.2)  (0 ≤) ∥T ∥ − w (T) ≤ \frac{1}{2} · \frac{r^{2}}{|λ|}.

Proof. For x ∈ H with ∥x∥ = 1, we have from (2.1) that

∥T x − λx∥ ≤ ∥T − λI∥ ≤ r,

giving

(2.3)  ∥T x∥^{2} + |λ|^{2} ≤ 2 Re [\overline{λ} ⟨T x, x⟩] + r^{2} ≤ 2 |λ| |⟨T x, x⟩| + r^{2}.

Taking the supremum over x ∈ H, ∥x∥ = 1 in (2.3) we get the following inequality that is of interest in itself:

(2.4)  ∥T ∥^{2} + |λ|^{2} ≤ 2w (T) |λ| + r^{2}.

Since, obviously,

(2.5)  ∥T ∥^{2} + |λ|^{2} ≥ 2 ∥T ∥ |λ| ,

hence by (2.4) and (2.5) we deduce the desired inequality (2.2).

Remark 1. If the operator T : H → H is such that R (T) ⊥ R (T^{∗}) , ∥T ∥ = 1 and ∥T − I∥ ≤ 1, then the equality case holds in (2.2). Indeed, by Theorem 6, we have in this case w (T) = \frac{1}{2} ∥T ∥ = \frac{1}{2} and since we can choose in Theorem 8, λ = 1, r = 1, then we get in both sides of (2.2) the same quantity \frac{1}{2}.

Problem 1. Find the bounded linear operators T : H → H with ∥T ∥ = 1, R (T) ⊥ R (T^{∗}) and ∥T − λI∥ ≤ |λ|^{\frac{1}{2}} .

The following corollary may be stated:

Corollary 1. Let A : H → H be a bounded linear operator and φ, ϕ ∈ ℂ with ϕ ≠ −φ, φ. If

(2.6)  Re ⟨ϕx − Ax, Ax − φx⟩ ≥ 0 for any x ∈ H, ∥x∥ = 1

then

(2.7)  (0 ≤) ∥A∥ − w (A) ≤ \frac{1}{4} · \frac{|ϕ − φ|^{2}}{|ϕ + φ|} .

Proof. Utilising the fact that in any Hilbert space the following two statements are equivalent:

(i) Re ⟨Z − x, x − z⟩ ≥ 0, x, z, Z ∈ H;

(ii) ∥x − \frac{z+Z}{2} ∥ ≤ \frac{1}{2} ∥Z − z∥ ,

we deduce that (2.6) is equivalent to

(2.8)  ∥Ax − \frac{ϕ + φ}{2} · Ix∥ ≤ \frac{1}{2} |ϕ − φ|

for any x ∈ H, ∥x∥ = 1, which in its turn is equivalent with the operator norm inequality:

(2.9)  ∥A − \frac{ϕ + φ}{2} · I∥ ≤ \frac{1}{2} |ϕ − φ| .

Now, applying Theorem 8 for T = A, λ = \frac{φ+ϕ}{2} and r = \frac{1}{2} |Γ − γ| , we deduce the desired result (2.7).

Remark 2. Following [1, p. 25], we say that an operator B : H → H is accreative, if Re ⟨Bx, x⟩ ≥ 0 for any x ∈ H. One may observe that the assumption (2.6) above is then equivalent with the fact that the operator (A^{∗} − φ̄I) (ϕI − A) is accreative.

Perhaps a more convenient sufficient condition in terms of positive operators is the following one:

Corollary 2. Let φ, ϕ ∈ ℂ with ϕ ≠ −φ, φ and A : H → H a bounded linear operator in H. If (A^{∗} − φ̄I) (ϕI − A) is self-adjoint and

(2.10)  (A^{∗} − φ̄I) (ϕI − A) ≥ 0

in the operator order, then

(2.11)  (0 ≤) ∥A∥ − w (A) ≤ \frac{1}{4} · \frac{|ϕ − φ|^{2}}{|ϕ + φ|} .

The following result may be stated as well:

Corollary 3. Assume that T, λ, r are as in Theorem 8. If, in addition, there exists ρ ≥ 0 such that

(2.12)  ||λ| − w (T)| ≥ ρ,

then

(2.13)  (0 ≤) ∥T ∥^{2} − w^{2} (T) ≤ r^{2} − ρ^{2}.

Proof. From (2.4) of Theorem 8, we have

(2.14)  ∥T ∥^{2} − w^{2} (T) ≤ r^{2} − w^{2} (T) + 2w (T) |λ| − |λ|^{2}
= r^{2} − (|λ| − w (T))^{2} .

On utilising (2.4) and (2.12) we deduce the desired inequality (2.13).

Remark 3. In particular, if ∥T − λI∥ ≤ r and |λ| = w (T) , λ ∈ ℂ, then

(2.15)  (0 ≤) ∥T ∥^{2} − w^{2} (T) ≤ r^{2}.

The following result may be stated as well.

Theorem 9. Let T : H → H be a nonzero bounded linear operator on H and λ ∈ ℂ \ {0} , r > 0 with |λ| > r. If

(2.16)  ∥T − λI∥ ≤ r,

INEQUALITIES FOR THE NUMERICAL RADIUS OF LINEAR OPERATORS 5

then

(2.17)  \sqrt{1 − \frac{r^{2}}{|λ|^{2}}} ≤ \frac{w (T)}{∥T ∥} (≤ 1) .

Proof. From (2.4) of Theorem 8, we have

∥T ∥^{2} + |λ|^{2} − r^{2} ≤ 2 |λ| w (T) ,

which implies, on dividing with \sqrt{|λ|^{2} − r^{2}} > 0 that

(2.18)  \begin{matrix}∥T ∥ & 2 \\ √ & \overline{|λ|^{2} − r^{2}} \end{matrix} + \sqrt{|λ|^{2} − r^{2}} ≤ \frac{2 |λ| w (T)}{\sqrt{|λ|^{2} − r^{2}}} .

By the elementary inequality

(2.19)  2 ∥T ∥ ≤ \begin{matrix}∥T ∥ & 2 \\ √ & \overline{|λ|^{2} − r^{2}} \end{matrix} + \sqrt{|λ|^{2} − r^{2}}

and by (2.18) we deduce

∥T ∥ ≤ \frac{w (T) |λ|}{\sqrt{|λ|^{2} − r^{2}}} ,

which is equivalent to (2.17). Remark 4. Squaring (2.17), we get the inequality

(2.20)  (0 ≤) ∥T ∥^{2} − w^{2} (T) ≤ \frac{r^{2}}{|λ|^{2}} ∥T ∥^{2} .

Remark 5. Since for any bounded linear operator T : H → H we have that w (T) ≥ \frac{1}{2} ∥T ∥ , hence (2.17) would produce a refinement of this classic fact only in the case when

\frac{1}{2} ≤ (1 − \frac{r^{2}}{|λ|^{2}} )^{\frac{1}{2}} ,

which is equivalent to r/ |λ| ≤ \sqrt{3}/2.

The following corollary holds. Corollary 4. Let φ, ϕ ∈ ℂ with Re (ϕφ̄) > 0. If T : H → H is a bounded linear operator such that either (2.6) or (2.10) holds true, then:

(2.21)  \frac{2\sqrt{Re (ϕφ̄)}}{|ϕ + φ|} ≤ \frac{w (T)}{∥T ∥} (≤ 1)

and

(2.22)  (0 ≤) ∥T ∥^{2} − w^{2} (T) ≤ |\frac{ϕ − φ}{ϕ + φ}|^{2} ∥T ∥^{2} .

Proof. If we consider λ = \frac{ϕ+φ}{2} and r = \frac{1}{2} |ϕ − φ| , then |λ|^{2}−r^{2} = |\frac{ϕ+φ}{2} |^{2}−|\frac{ϕ−φ}{2} |^{2} = Re (ϕφ̄) > 0. Now, on applying Theorem 9, we deduce the desired result. Remark 6. If |ϕ − φ| ≤ \frac{\sqrt{3}}{2} |ϕ + φ| , Re (ϕφ̄) > 0, then (2.21) is a refinement of the inequality w (T) ≥ \frac{1}{2} ∥T ∥ .

The following result may be of interest as well.

Theorem 10. Let T : H → H be a nonzero bounded linear operator on H and λ ∈ ℂ\ {0} , r > 0 with |λ| > r. If

(2.23)  ∥T − λI∥ ≤ r,

then

(2.24)  (0 ≤) ∥T ∥^{2} − w^{2} (T) ≤ \frac{2r^{2}}{|λ| + \sqrt{|λ|^{2} − r^{2}}} w (T) .

Proof. From the proof of Theorem 8, we have

(2.25)  ∥T x∥^{2} + |λ|^{2} ≤ 2 Re [\overline{λ} ⟨T x, x⟩] + r^{2}

for any x ∈ H, ∥x∥ = 1.

If we divide (2.25) by |λ| |⟨T x, x⟩| , (which, by (2.25), is positive) then we obtain

(2.26)  \begin{matrix}∥T x∥ & 2 \\ |λ| & |⟨T x, x⟩| \end{matrix} ≤ \frac{2 Re [\overline{λ} ⟨T x, x⟩]}{|λ| |⟨T x, x⟩|} + \frac{r^{2}}{|λ| |⟨T x, x⟩|} − \begin{matrix}| & λ & | \\ | & ⟨T x, x⟩ & | \end{matrix}

for any x ∈ H, ∥x∥ = 1.

If we subtract in (2.26) the same quantity \begin{matrix}| & ⟨T x,x⟩ & | \\ | & λ & | \end{matrix} from both sides, then we get

\begin{matrix}∥T x∥ & 2 \\ |λ| & |⟨T x, x⟩| \end{matrix} − \begin{matrix}| & ⟨T x, x⟩ & | \\ | & λ & | \end{matrix} (2.27)

≤ \frac{2 Re [\overline{λ} ⟨T x, x⟩]}{|λ| |⟨T x, x⟩|} + \frac{r^{2}}{|λ| |⟨T x, x⟩|} − \begin{matrix}| & ⟨T x, x⟩ & | \\ | & λ & | \end{matrix} − \begin{matrix}| & λ & | \\ | & ⟨T x, x⟩ & | \end{matrix}

= \frac{2 Re [\overline{λ} ⟨T x, x⟩]}{|λ| |⟨T x, x⟩|} − \frac{|λ|^{2} − r^{2}}{|λ| |⟨T x, x⟩|} − \begin{matrix}| & ⟨T x, x⟩ & | \\ | & λ & | \end{matrix}

= \frac{2 Re [\overline{λ} ⟨T x, x⟩]}{|λ| |⟨T x, x⟩|} − (\begin{matrix}√ & \overline{|λ|^{2} − r^{2}} \\ √ & \overline{|λ| |⟨T x, x⟩|} \end{matrix} − \begin{matrix}√ & \overline{|⟨T x, x⟩|} \\ √ & \overline{|λ|} \end{matrix})^{2} − 2\begin{matrix}√ & \overline{|λ|^{2} − r^{2}} \\ | & λ & | \end{matrix} .

Since

Re [\overline{λ} ⟨T x, x⟩] ≤ |λ| |⟨T x, x⟩|

and

(\begin{matrix}√ & \overline{|λ|^{2} − r^{2}} \\ √ & \overline{|λ| |⟨T x, x⟩|} \end{matrix} − \begin{matrix}√ & \overline{|⟨T x, x⟩|} \\ √ & \overline{|λ|} \end{matrix})^{2} ≥ 0

hence by (2.27) we get

\begin{matrix}∥T x∥ & 2 \\ |λ| & |⟨T x, x⟩| \end{matrix} − \begin{matrix}| & ⟨T x, x⟩ & | \\ | & λ & | \end{matrix} ≤ \frac{2 (|λ| − \sqrt{|λ|^{2} − r^{2}})}{|λ|}

which gives the inequality

(2.28)  ∥T x∥^{2} ≤ |⟨T x, x⟩|^{2} + 2 |⟨T x, x⟩| (|λ| − \sqrt{|λ|^{2} − r^{2}})

for any x ∈ H, ∥x∥ = 1.

Taking the supremum over x ∈ H, ∥x∥ = 1, we get

∥T ∥^{2} ≤ sup \{|⟨T x, x⟩|^{2} + 2 |⟨T x, x⟩| (|λ| − \sqrt{|λ|^{2} − r^{2}})\}

≤ sup \{|⟨T x, x⟩|^{2}\} + 2 (|λ| − \sqrt{|λ|^{2} − r^{2}}) sup \{|⟨T x, x⟩|\}

= w^{2} (T) + 2 (|λ| − \sqrt{|λ|^{2} − r^{2}}) w (T) ,

which is clearly equivalent to (2.24).

Corollary 5. Let φ, ϕ ∈ ℂ with Re (ϕφ̄) > 0. If A : H → H is a bounded linear operator such that either (2.6) or (2.10) hold true, then:

(2.29)  (0 ≤) ∥A∥^{2} − w^{2} (A) ≤ [|ϕ + φ| − 2\sqrt{Re (ϕφ̄)}] w (A) .

Remark 7. If M ≥ m > 0 are such that either (A^{∗} − mI) (MI − A) is accreative, or, sufficiently, (A^{∗} − mI) (MI − A) is self-adjoint and

(2.30)  (A^{∗} − mI) (MI − A) ≥ 0 in the operator order,

then, by (2.21) we have:

(2.31)  (1 ≤) \frac{∥A∥}{w (A)} ≤ \frac{M + m}{2\sqrt{mM}} ,

which is equivalent to

(2.32)  (0 ≤) ∥A∥ − w (A) ≤ \frac{(\sqrt{M} − \sqrt{m})^{2}}{2\sqrt{mM}} w (A) ,

while from (2.24) we have

(2.33)  (0 ≤) ∥A∥^{2} − w^{2} (A) ≤ (\sqrt{M} − \sqrt{m})^{2} w (A) .

Also, the inequality (2.7) becomes

(2.34)  (0 ≤) ∥A∥ − w (A) ≤ \frac{1}{4} · \frac{(M − m)^{2}}{M + m} .

References

[1] K.E. GUSTAFSON and D.K.M. RAO, Numerical Range, Springer-Verlag, New York, Inc., 1997.

[2] P.R. HALMOS, A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, Berlin, Second edition, 1982.

School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne City, Victoria 8001, Australia.

   E-mail address: sever.dragomir@vu.edu.au
   URL: http://rgmia.vu.edu.au/dragomir