SOME INEQUALITIES FOR NORMAL OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Some inequalities for normal operators in Hilbert spaces are given. For this purpose, some results for vectors in inner product spaces due to Buzano, Dunkl-Williams, Hile, Goldstein-Ryff-Clarke, Dragomir-Sándor and the author are employed.

1. Introduction

Let (H; ⟨·, ·⟩) be a complex Hilbert space and T : H → H a bounded linear operator on H. Recall that T is a normal operator if T^{∗}T = T T^{∗}. Normal operators may be regarded as a generalisation of self-adjoint operator T in which T^{∗} need not be exactly T but commutes with T [8, p. 15].

The numerical range of an operator T is the subset of the complex numbers ℂ given by [8, p. 1]:

W (T) = {⟨T x, x⟩ , x ∈ H, ∥x∥ = 1} .

For various properties of the numerical range see [8].

We recall here some of the ones related to normal operators.

Theorem 1. If W (T) is a line segment, then T is normal.

We denote by r (T) the operator spectral radius [8, p. 10] and by w (T) its numerical radius [8, p. 8]. The following result may be stated as well [8, p. 15].

Theorem 2. If T is normal, then ∥T^{n}∥ = ∥T ∥^{n} , n = 1, …. Moreover, we have:

(1.1)  r (T) = w (T) = ∥T ∥ .

An important fact about the normal operators that will be used frequently in the sequel is the following one [9, p. 42]:

Theorem 3. A necessary and sufficient condition that an operator T be normal is that ∥T x∥ = ∥T^{∗}x∥ for every vector x ∈ H.

The aim of this paper is to establish some inequalities for normal operators in Hilbert spaces. For this purpose, some inequalities for vectors in inner product spaces due to Buzano, Dunkl-Williams, Hile, Goldstein-Ryff-Clarke, Dragomir-Sándor and the author are employed.

Date: 30 August, 2005.

2000 Mathematics Subject Classification. 47A12.

Key words and phrases. Numerical radius, Bounded linear operators, Normal operators, Hilbert spaces, Norm inequalities.

2. Inequalities for Vectors

The following result may be stated.

Theorem 4. Let (H; ⟨·, ·⟩) be a Hilbert space and T : H → H a normal linear operator on H. Then

(2.1)  \frac{1}{2} (∥T x∥^{2} + |⟨T^{2}x, x⟩|) ≥ |⟨T x, x⟩|^{2} ,

for any x ∈ H, ∥x∥ = 1.The constant \frac{1}{2} is best possible in (2.1).

Proof. We need the following refinement of Schwarz’s inequality obtained by the author in 1985 [2, Theorem 2] (see also [5] and [4]):

(2.2)  ∥a∥ ∥b∥ ≥ |⟨a, b⟩ − ⟨a, e⟩ ⟨e, b⟩| + |⟨a, e⟩ ⟨e, b⟩| ≥ |⟨a, b⟩| ,

provided a, b, e are vectors in H and ∥e∥ = 1.

Observing that

|⟨a, b⟩ − ⟨a, e⟩ ⟨e, b⟩| ≥ |⟨a, e⟩ ⟨e, b⟩| − |⟨a, b⟩| ,

then by the first inequality in (2.2) we deduce

(2.3)  \frac{1}{2} (∥a∥ ∥b∥ + |⟨a, b⟩|) ≥ |⟨a, e⟩ ⟨e, b⟩| .

This inequality was obtained in a different way earlier by M.L. Buzano in [1]. Now, choose in (2.3), e = x, ∥x∥ = 1, a = T x and b = T^{∗}x to get

(2.4)  \frac{1}{2} (∥T x∥ ∥T^{∗}x∥ + |⟨T^{2}x, x⟩|) ≥ |⟨T x, x⟩|^{2}

for any x ∈ H, ∥x∥ = 1. Since T is normal, then ∥T x∥ = ∥T^{∗}x∥ , and by (2.4) we deduce the desire result (2.1).

The fact that, the constant \frac{1}{2} is best possible in (2.1) is obvious since for T = I, the identity operator, we get equality in (2.1).

From a different perspective, we can state the followin reverse result:

Theorem 5. Let T : H → H be a normal operator on the Hilbert space (H; ⟨·, ·⟩) . If λ ∈ ℂ, then

(2.5)  (0 ≤) ∥T x∥^{2} − |⟨T^{2}x, x⟩| ≤ \frac{2}{(1 + |λ|)^{2}} ∥T x − λT^{∗}x∥^{2}

for any x ∈ H, ∥x∥ = 1.

Proof. We use the following inequality [6]:

∥a − b∥ ≥ \frac{1}{2} (∥a∥ + ∥b∥) ∥\frac{a}{∥a∥} − \frac{b}{∥b∥}∥ , a, b ∈ H\ \{0\} ,

which is well known in the literature as the Dunkl-Williams inequality. This inequality, by taking the square, is clearly equivalent to

\frac{4 ∥a − b∥^{2}}{(∥a∥ + ∥b∥)^{2}} ≥ ∥\frac{a}{∥a∥} − \frac{b}{∥b∥}∥^{2} = 2 − 2 · \begin{matrix}Re & ⟨a, b⟩ \\ ∥a∥ & ∥b∥ \end{matrix},

which shows that (see [3, Eq. (2.5)])

\frac{∥a∥ ∥b∥ − |⟨a, b⟩|}{∥a∥ ∥b∥} ≤ \frac{2 ∥a − b∥^{2}}{(∥a∥ + ∥b∥)^{2}} .

Now, for x ∈ H\ ker (T) , ∥x∥ = 1, choose a = T x and b = λT^{∗}x (λ ≠ 0) to obtain

(2.6)  ∥T x∥ ∥T^{∗}x∥ − |⟨T^{2}x, x⟩| ≤ \frac{2 ∥T x∥ ∥T^{∗}x∥}{(∥T x∥ + |λ| ∥T^{∗}x∥)^{2}} ∥T x − λT^{∗}x∥^{2} .

Since ∥T x∥ = ∥T^{∗}x∥ , T being a normal operator, we get from (2.6) that (2.5) holds true for any x ∈ H\ ker (T) , ∥x∥ = 1.

For λ = 0 the inequality (2.5) is obvious.

Since for normal operators ker (T) = ker (T^{∗}) then for x ∈ ker (T) , ∥x∥ = 1 the inequality (2.5) also holds true.

The following result which provides a different bound for the nonnegative quantity

∥T x∥^{2} − |⟨T^{2}x, x⟩| , x ∈ H, ∥x∥ = 1

may be stated as well:

Theorem 6. Let T : H → H be a normal operator on the Hilbert space H and α, λ ∈ ℂ\ {0} . Then

(0 ≤) ∥T x∥^{2} − |⟨T^{2}x, x⟩| (2.7)

≤ \frac{1}{2} · \begin{matrix}[|Re α| ∥T x − \frac{α}{ᾱ}λT^{∗}x∥ + |Im α| ∥T x + \frac{α}{ᾱ}λT^{∗}x∥] & 2 \\ |λ| & |α| & 2 \end{matrix}

for any x ∈ H, ∥x∥ = 1.

Proof. We use the following inequality (see [3, Theorem 2.11]):

(2.8)  ∥a∥ ∥b∥ − Re [\begin{matrix}α & 2 \\ |α| & 2 \end{matrix} ⟨a, b⟩] ≤ \frac{1}{2} · \begin{matrix}[|Re α| ∥a − b∥ + |Im α| ∥a + b∥] & 2 \\ |α| & 2 \end{matrix}

for the choices:

a = \frac{T x}{α}, b = \frac{λ}{ᾱ}T^{∗}x, x ∈ H

to obtain:

\begin{matrix}|λ| & ∥T x∥ & ∥T^{∗}x∥ \\ |α| & 2 \end{matrix} − Re [\begin{matrix}α & 2 \\ |α| & 2 \end{matrix} · \frac{λ̄}{α^{2}} ⟨T x, T^{∗}x⟩] (2.9)

≤ \frac{1}{2} · \begin{matrix}[|Re α| ∥\frac{T x}{α} − \frac{λ}{α}T^{∗}x∥ + |Im α| ∥\frac{T x}{α} + \frac{λ}{α}T^{∗}x∥] & 2 \\ |α| & 2 \end{matrix} .

Since T is normal, we get from (2.9) the desired result (2.7). The details are omitted.

Another result of this type is incorporated in:

Theorem 7. Let T : H → H be a normal operator on the Hilbert space H, s ∈ [0, 1] and t ∈ ℝ. Then

(0 ≤) ∥T x∥^{4} − |⟨T^{2}x, x⟩|^{2} (2.10)

≤ ∥T x∥^{2} [s ∥tT^{∗}x − T x∥^{2} + (1 − s) ∥T^{∗}x − tT x∥^{2}] .

4 S.S. DRAGOMIR

In particular

(0 ≤) ∥T x∥^{4} − |⟨T^{2}x, x⟩|^{2}
≤ \frac{1}{2} ∥T x∥^{2} inf_{t∈ℝ} [∥tT^{∗}x − T x∥^{2} + ∥T^{∗}x − tT x∥^{2}] .

Proof. We use the inequality obtained in [4, Theorem 2], to state that [(1 − s) ∥a∥^{2} + s ∥b∥^{2}] [(1 − s) ∥b∥^{2} + s ∥a∥^{2}] − |⟨a, b⟩|^{2} (2.11)

≤ [(1 − s) ∥a∥^{2} + s ∥b∥^{2}] [(1 − s) ∥b − ta∥^{2} + s ∥tb − a∥^{2}]

for any s ∈ [0, 1], t ∈ ℝ and a, b ∈ H.

If in (2.11) we choose a = T x, b = T^{∗}x, x ∈ H and ∥x∥ = 1, then we get

∥T x∥^{4} − |⟨T^{2}x, x⟩|^{2}

≤ ∥T x∥^{2} [s ∥tT^{∗}x − T x∥^{2} + (1 − s) ∥T^{∗}x − tT x∥^{2}]

for any s ∈ [0, 1], t ∈ ℝ, from where we deduce the desired inequality (2.10).

From a different perspective, we can state the following result as well. Theorem 8. Let T : H → H be a normal operator on the Hilbert space (H; ⟨·, ·⟩) . If λ ∈ ℂ\ {0} and r > 0 are such that

(2.12)  ∥T − λT^{∗}∥ ≤ r,

then:

(2.13)  (0 ≤) ∥T x∥^{4} − |⟨T^{2}x, x⟩|^{2} ≤ \frac{r^{2}}{|λ|^{2}} ∥T x∥^{2}

for any x ∈ H, ∥x∥ = 1. Proof. We use the following reverse of the quadratic Schwarz inequality obtained by the author in [4]

(2.14)  (0 ≤) ∥a∥^{2} ∥b∥^{2} − |⟨a, b⟩|^{2} ≤ \frac{1}{|α|^{2}} ∥a∥^{2} ∥a − αb∥^{2}

provided a, b ∈ H and α ∈ ℂ\ {0} .

Choosing in (2.14) a = T x, α = λ, b = T^{∗}x, we get

(2.15)  ∥T x∥^{4} ≤ |⟨T^{2}x, x⟩|^{2} + \frac{1}{|λ|^{2}} ∥T x∥^{2} ∥T x − λT^{∗}x∥^{2}

≤ |⟨T^{2}x, x⟩|^{2} + \frac{1}{|λ|^{2}} r^{2} ∥T x∥^{2}

which is the desired result (2.13).

Finally, on utilising the following result obtained in [4]: Lemma 1. Let a, b ∈ H\ \{0\} and ε ∈ (0, \frac{1}{2}] . If

(2.16)  (0 ≤) 1 − ε − \sqrt{1 − 2ε} ≤ \frac{∥a∥}{∥b∥} ≤ 1 − ε + \sqrt{1 − 2ε},

then

(2.17)  (0 ≤) ∥a∥ ∥b∥ − Re ⟨a, b⟩ ≤ ε ∥a − b∥^{2} ,

we can state:

Theorem 9. Let T : H → H be a normal operator on H. If λ ∈ ℂ is such that

(2.18)  (0 ≤) 1 − ε − \sqrt{1 − 2ε} ≤ |λ| ≤ 1 − ε + \sqrt{1 − 2ε}, ε ∈ (0, \frac{1}{2}]

then

(2.19)  (0 ≤) ∥T x∥^{2} − |⟨T^{2}x, x⟩| ≤ \frac{ε}{|λ|} ∥T x − λT^{∗}x∥^{2}

for any x ∈ H, ∥x∥ = 1.

Proof. Utilising Lemma 1 for a = λT^{∗}x, b = T x, x ∈ H\ ker (T) , ∥x∥ = 1, we have

(2.20)  |λ| ∥T x∥^{2} − |λ| |⟨T^{2}x, x⟩| ≤ ε ∥T x − λT^{∗}x∥^{2} .

For x ∈ ker (T) , ∥x∥ = 1 the inequality (2.19) also holds, and the proof is completed.

3. Inequalities for Operator Norm

The following result concerning the operator norm can be stated:

Theorem 10. Let T : H → H be a normal operator on the Hilbert space H. If λ ∈ ℂ, |λ| ≠ 1, then:

(3.1)  ∥T − |λ|^{v+1} T^{∗}∥ ≤ \frac{1 − |λ|^{v+1}}{1 − |λ|} ∥T − λT^{∗}∥ ,

for any v > 0.

Proof. We use the following inequality:

(3.2)  ∥∥a∥^{v} a − ∥b∥^{v} b∥ ≤ \frac{∥a∥^{v+1} − ∥b∥^{v+1}}{∥a∥ − ∥b∥} ∥a − b∥

provided v > 0 and ∥a∥ ≠ ∥b∥ , which is known in the literature as the Hile inequality [10].

Now, if we choose in (3.2) a = T x, b = λT^{∗}x, since T is normal, we have ∥a∥ = ∥T x∥ , ∥b∥ = |λ| ∥T x∥ and by (3.2) we get

(3.3)  ∥T x∥^{v} ∥T x − |λ|^{v+1} T^{∗}x∥ ≤ ∥T x∥^{v} \frac{(1 − |λ|^{v+1})}{1 − |λ|} ∥T x − λT^{∗}x∥

for any x ∈ H \ ker (T) .

If x ∉ ker (T) , then from (3.3) we get

(3.4)  ∥T x − |λ|^{v+1} T^{∗}x∥ ≤ \frac{1 − |λ|^{v+1}}{1 − |λ|} ∥T x − λT^{∗}x∥ .

If x ∈ ker (T) and since ker (T) = ker (T^{∗}) , T being normal, then the inequality (3.4) is also valid. Therefore, (3.4) holds for any x ∈ H.

Taking the supremum over x ∈ H, ∥x∥ = 1, we get the desired inequality (3.1).

Remark 1. For v = 1, we get the inequality:

(3.5)  ∥T − |λ|^{2} T^{∗}∥ ≤ (1 + |λ|) ∥T − λT^{∗}∥ .

Utilising the second inequality due to Hile (see [10, Eq. (5.2)]):

∥\frac{a}{∥a∥^{v+2}} − \frac{b}{∥b∥^{v+2}} ∥ ≤ \frac{∥a∥^{v+2} − ∥b∥^{v+2}}{∥a∥ − ∥b∥} · \frac{∥a − b∥}{∥a∥^{v+1} · ∥b∥^{v+1}}

for a, b ∈ H, a, b ≠ 0 and ∥a∥ ≠ ∥b∥ , and making use of an argument similar to the one in the proof of the above theorem, we can state the following result:

Theorem 11. Let T : H → H be a normal operator on the Hilbert space H. If λ ∈ ℂ, |λ| ≠ 0, 1, then:

(3.6)  ∥T − \frac{λ}{|λ|^{v+2}} T^{∗}∥ ≤ \frac{1 − |λ|^{v+1}}{(1 − |λ|) |λ|^{v+1}} ∥T − λT^{∗}∥ ,

where v > 0.

The following result may be stated as well.

Theorem 12. Let T : H → H be a normal operator on the Hilbert space H. If |λ| ≤ 1, then

(3.7)  (1 − |λ|^{ρ})^{2} ∥T ∥^{2} ≤ \{ \begin{matrix}ρ^{2} ∥T − λT^{∗}∥^{2} & if ρ ≥ 1, \\ |λ|^{2ρ−2} ∥T − λT^{∗}∥^{2} & if ρ < 1. \end{matrix}

Proof. We use the following inequality due to Goldstein, Ryff and Clarke [7]

(3.8) ∥a∥^{2ρ} + ∥b∥^{2ρ} − 2 ∥a∥^{ρ−1} ∥b∥^{ρ−1} Re ⟨a, b⟩

≤ \{ \begin{matrix}ρ^{2} ∥a∥^{2ρ−2} ∥a − b∥^{2} & if ρ ≥ 1, \\ ∥b∥^{2ρ−2} ∥a − b∥^{2} & if ρ < 1, \end{matrix}

provided ρ ∈ ℝ and a, b ∈ H with ∥a∥ ≥ ∥b∥ .

Since Re ⟨a, b⟩ ≤ |⟨a, b⟩| , then, from (3.8), we have the inequality

(3.9) ∥a∥^{2ρ} + ∥b∥^{2ρ} ≤ 2 ∥a∥^{ρ−1} ∥b∥^{ρ−1} |⟨a, b⟩|

+ \{ \begin{matrix}ρ^{2} ∥a∥^{2ρ−2} ∥a − b∥^{2} & if ρ ≥ 1, \\ ∥b∥^{2ρ−2} ∥a − b∥^{2} & if ρ < 1. \end{matrix}

We choose a = T x, b = λT^{∗}x and since |λ| ≤ 1, we have ∥a∥ ≥ ∥b∥ . From (3.9), on taking into account that ∥T x∥ = ∥T^{∗}x∥ , we deduce

∥T x∥^{2ρ} + |λ|^{2ρ} ∥T x∥^{2ρ} ≤ 2 ∥T x∥^{2ρ−2} |λ|^{ρ} |⟨T^{2}x, x⟩|

+ \{ \begin{matrix}ρ^{2} ∥T x∥^{2ρ−2} ∥T x − λT^{∗}x∥^{2} & if ρ ≥ 1, \\ |λ|^{2ρ−2} ∥T x∥^{2ρ−2} ∥T x − λT^{∗}x∥^{2} & if ρ < 1, \end{matrix}

which implies that:

(3.10) (1 + |λ|^{2ρ}) ∥T x∥^{2}

≤ 2 |λ|^{ρ} |⟨T^{2}x, x⟩| + \{ \begin{matrix}ρ^{2} ∥T x − λT^{∗}x∥^{2} & if ρ ≥ 1, \\ |λ|^{2ρ−2} ∥T x − λT^{∗}x∥^{2} & if ρ < 1, \end{matrix}

for any x ∈ H, ∥x∥ = 1.

This inequality is of interest in itself.

Taking the supremum over x ∈ H, ∥x∥ = 1, and using the fact that

sup_{∥x∥=1} |⟨T^{2}x, x⟩| = w (T^{2}) = ∥T ∥^{2} ,

we get the desired inequality (3.7).

Remark 2. If |λ| > 1, on choosing in (3.9) a = λT^{∗}x, b = T x we get:

(|λ|^{2ρ} + 1) ∥T x∥^{2} ≤ 2 |λ|^{ρ} |⟨T^{2}x, x⟩|

+ \{ \begin{matrix}ρ^{2} |λ|^{2ρ−2} ∥T x − λT^{∗}x∥^{2} & if ρ ≥ 1, \\ ∥T x − λT^{∗}x∥^{2} & if ρ < 1, \end{matrix}

which implies the “dual” inequality:

(3.11)  (1 − |λ|^{ρ})^{2} ∥T ∥^{2} ≤ \{ \begin{matrix}ρ^{2} |λ|^{2ρ−2} ∥T − λT^{∗}∥^{2} & if ρ ≥ 1, \\ ∥T − λT^{∗}∥^{2} & if ρ < 1, \end{matrix}

for any λ ∈ ℂ, |λ| > 1.

The following result concerning operator norm inequalities may be stated as well:

Theorem 13. Let T : H → H be a normal operator on the Hilbert space (H; ⟨·, ·⟩) and α, β ∈ ℂ. Then:

(3.12)  ∥T ∥^{p} [(|α| + |β|)^{p} + ||α| − |β||^{p}] ≤ ∥αT + βT^{∗}∥^{p} + ∥αT − βT^{∗}∥^{p}

if p ∈ (1, 2) and

(3.13)  ∥αT + βT^{∗}∥^{p} + ∥αT − βT^{∗}∥^{p} ≥ 2 (|α|^{p} + |β|^{p}) ∥T ∥^{p}

if p ≥ 2.

Proof. We use the following result obtained by Dragomir and Sándor in [5]:

(3.14)  ∥a + b∥^{p} + ∥a − b∥^{p} ≥ (∥a∥ + ∥b∥)^{p} + |∥a∥ − ∥b∥|^{p}

if p ∈ (1, 2) and

(3.15)  ∥a + b∥^{p} + ∥a − b∥^{p} ≥ 2 (∥a∥^{p} + ∥b∥^{p})

if p ≥ 2, where a, b are arbitrary vectors in the inner product space (H; ⟨·, ·⟩) . We choose a = αT x, b = βT^{∗}x to get:

∥(αT + βT^{∗}) (x)∥^{p} + ∥(αT − βT^{∗}) (x)∥^{p} (3.16)

≥ (|α| + |β|)^{p} ∥T x∥^{p} + ||α| − |β||^{p} ∥T x∥^{p}
= [(|α| + |β|)^{p} + ||α| − |β||^{p}] ∥T x∥^{p}

if p ∈ (1, 2) and

(3.17)  ∥(αT + βT^{∗}) (x)∥^{p} + ∥(αT − βT^{∗}) (x)∥^{p} ≥ 2 (|α|^{p} + |β|^{p}) ∥T x∥^{p}

if p ≥ 2.

Taking the supremum over x ∈ H, ∥x∥ = 1, we deduce (3.12) and (3.13).

Remark 3. The case p = 2 produces the following inequality:

∥αT + βT^{∗}∥^{2} + ∥αT − βT^{∗}∥^{2} ≥ 2 (|α|^{2} + |β|^{2}) ∥T ∥^{2} ,

that can also be obtained by utilising the parallelogram identity.

The following general result may be stated as well:

Theorem 14. Let T : H → H be a normal operator on the Hilbert space H. If α, β ∈ ℂ and r, ρ > 0 are such that

(3.18)  ∥T − ᾱI∥ ≤ r and ∥T^{∗} − βI∥ ≤ ρ,

then

(3.19)  ∥T ∥^{2} + \frac{1}{2} (|α|^{2} + |β|^{2}) ≤ \frac{1}{2} (r^{2} + ρ^{2}) + ∥αT + βT^{∗}∥ .

Proof. The condition (3.18) obviously implies that

(3.20)  ∥T x∥^{2} + |α|^{2} ≤ 2 Re ⟨(αT) x, x⟩ + r^{2}

and

(3.21)  ∥T^{∗}x∥^{2} + |β|^{2} ≤ 2 Re ⟨(βT)^{∗} x, x⟩ + ρ^{2}

for any x ∈ H, ∥x∥ = 1.

Adding (3.20) and (3.21) and taking into account that ∥T x∥ = ∥T^{∗}x∥, we obtain

(3.22)  2 ∥T x∥^{2} + |α|^{2} + |β|^{2} ≤ 2 Re ⟨(αT + βT^{∗}) x, x⟩ + r^{2} + ρ^{2}
≤ 2 |⟨(αT + βT^{∗}) x, x⟩| + r^{2} + ρ^{2}.

Taking the supremum on (3.22) over x ∈ H, ∥x∥ = 1, and utilising the fact that for the normal operator T we have

w (αT + βT^{∗}) = ∥αT + βT^{∗}∥

then we get the desired inequality (3.19).

Remark 4. If α, β ∈ ℂ and r, ρ > 0 are such that |α|^{2} + |β|^{2} = ρ^{2} + r^{2}, then from (3.19) we have:

(3.23)  ∥T ∥^{2} ≤ ∥αT + βT^{∗}∥ .

References

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[2] S.S. DRAGOMIR, Some refinements of Schwarz inequality, Simposional de Matematică şi Aplicaţii, Polytechnical Institute Timişoara, Romania, 1-2 Nov., 1985, 13-16. ZBL 0594:46018.

[3] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner product spaces (I), J. Ineq. Pure & Appl. Math., 6(3) (2005), Art. 59. [ONLINE http://jipam.vu.edu.au/article.php?sid=532].

[4] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner product spaces (II), RGMIA Res. Rep. Coll., 8(Supp) (2005), Art. 5. [ONLINE http://rgmia.vu.edu.au/v8(E).html].

[5] S.S. DRAGOMIR and J. SÁNDOR, Some inequalities in prehilbertian spaces, Studia Univ. “Babeş-Bolyai”- Mathematica, 32(1) (1987), 71-78.

[6] C.F. DUNKL and K.S. WILLIAMS, A simple norm inequality, Amer. Math. Monthly, 71(1) (1964), 43-44.

[7] A.A. GOLDSTEIN, J.V. RYFF and L.E. CLARKE, Problem 5473, Amer. Math. Monthly, 75(3) (1968), 309.

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

[9] P.R. HALMOS, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea Pub. Comp, New York, N.Y., 1972.

[10] G.H. HILE, Entire solutions of linear elliptic equations with Laplacian principal part, Pacific J. Math., 62(1) (1976), 127-140.

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