SOME INEQUALITIES FOR (α, β)-NORMAL OPERATORS IN HILBERT SPACES

SEVER S. DRAGOMIR^{1} AND MOHAMMAD SAL MOSLEHIAN^{2}

Abstract. An operator T is called (α, β)-normal (0 ≤ α ≤ 1 ≤ β) if

α^{2}T^{∗}T ≤ T T^{∗} ≤ β^{2}T^{∗}T.

In this paper, we establish various inequalities between the operator norm and its numerical radius of (α, β)-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.

1. Introduction

An operator T acting on a Hilbert space (H; ⟨·, ·⟩) is called (α, β)-normal (0 ≤ α ≤ 1 ≤ β) if

α^{2}T^{∗}T ≤ T T^{∗} ≤ β^{2}T^{∗}T.

Then

α^{2}⟨T^{∗}T x, x⟩ ≤ ⟨T T^{∗}x, x⟩ ≤ β^{2}⟨T^{∗}T x, x⟩ ,

whence

(1.1)  α∥T x∥ ≤ ∥T^{∗}x∥ ≤ β∥T x∥,

for all x ∈ H. If T is invertible, then so is the bounded operator T^{∗}T^{−1}. Hence T^{∗}T^{−1} is bounded below and so T is (α, β)-normal for some α and β. Normal and hyponormal operators are trivially (α, β)-normal for some appropriate values of α and β. There are however operators which are neither normal nor hyponormal. The following example of an (α, β)-normal with α = \sqrt{\frac{3−\sqrt{5}}{2}} and

β = \sqrt{\frac{3+\sqrt{5}}{2}} is due to M. Mirzavaziri

[ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] ∈ B(ℂ^{2}).

2000 Mathematics Subject Classification. 47A12.

Key words and phrases. Numerical radius, Bounded linear operator, Hilbert space, (α, β)- normal operator.

Let (H; ⟨·, ·⟩) be a complex Hilbert space. The numerical radius w(T) of an operator T on H is given by

(1.2)  w(T) = sup{|⟨T x, x⟩|, ∥x∥ = 1}.

Obviously, by (1.2), for any x ∈ H one has

(1.3)  |⟨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. Moreover, we have

w(T) ≤ ∥T ∥ ≤ 2w(T) (T ∈ B(H)).

For other results and historical comments on the numerical radius see [10].

In this paper, we establish various inequalities between the operator norm and its numerical radius of (α, β)-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Dunkl–Williams, Dragomir–Sándor, Goldstein–Ryff–Clarke and Dragomir.

2. Inequalities Involving Numerical Radius

In this section we study some inequalities concerning the numerical radius and norm of (α, β)-normal operators. Our first result reads as follows, see also [6]:

Theorem 2.1. Let T ∈ B(H) be an (α, β)-normal operator. Then

(2.1) (α^{2r} + β^{2r})∥T ∥^{2} ≤ \{ \begin{matrix}2β^{r}w(T^{2}) + r^{2}β^{2r−2}∥βT − T^{∗}∥^{2}, & if r ≥ 1, \\ 2β^{r}w(T^{2}) + ∥βT − T^{∗}∥^{2}, & if r < 1. \end{matrix}

Proof. We use the following inequality for vectors in inner product spaces due to Goldstein, Ryff and Clarke [9]:

(2.2) ∥a∥^{2r} + ∥b∥^{2r} − 2∥a∥^{r}∥b∥^{r} · \frac{Re⟨a, b⟩}{∥a∥ ∥b∥} ≤ \{ \begin{matrix}r^{2}∥a∥^{2r−2}∥a − b∥^{2} & if r ≥ 1, \\ ∥b∥^{2r−2}∥a − b∥^{2} & if r < 1, \end{matrix}

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

Suppose that r ≥ 1. Let x ∈ H with ∥x∥ = 1. Noting to (1.1) and applying (2.2) for the choices a = βT x, b = T^{∗}x we get

(2.3) ∥βT x∥^{2r} + ∥T^{∗}x∥^{2r} − 2∥βT x∥^{r−1}∥ ∥T^{∗}x∥^{r−1} Re⟨βT x, T^{∗}x⟩ ≤ r^{2}∥βT x∥^{2r−2}∥βT x − T^{∗}x∥^{2} for any x ∈ H, ∥x∥ = 1 and r ≥ 1. Using (1.1) and (2.3) we get

(2.4) (α^{2r} + β^{2r})∥T x∥^{2r} ≤ 2β^{r}∥T x∥^{r−1}∥T^{∗}x∥^{r−1}|⟨T^{2}x, x⟩| + r^{2}β^{2r−2}∥T x∥^{2r−2}∥βT x − T^{∗}x∥^{2}.

Taking the supremum in (2.4) over x ∈ H, ∥x∥ = 1, we deduce

(α^{2r} + β^{2r})∥T ∥^{2r} ≤ 2β^{r}∥T ∥^{2r−2}∥T^{∗}∥^{r−1}w(T^{2}) + r^{2}β^{2r−2}∥T ∥^{2r−2}∥βT − T^{∗}∥^{2},

which is the first inequality in (2.1). If r < 1, then one can similarly prove the second inequality in (2.1).

Theorem 2.2. Let T ∈ B(H) be an (α, β)-normal operator. Then

(2.5)  w(T)^{2} ≤ \frac{1}{2} [β∥T ∥^{2} + w(T^{2})] .

Proof. The following inequality is known in the literature as the Buzano inequality [1]:

(2.6)  |⟨a, e⟩⟨e, b⟩| ≤ \frac{1}{2}(∥a∥ ∥b∥ + |⟨a, b⟩|),

for any a, b, e in H with ∥e∥ = 1.

Let x ∈ H with ∥x|| = 1. Put e = x, a = T x, b = T^{∗}x in (2.6) to get

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

≤ \frac{1}{2}(β∥T x∥^{2} + |⟨T^{2}x, x⟩|).

Taking the supremum over x ∈ H, ∥x∥ = 1, we obtain (2.5).

Theorem 2.3. Let T ∈ B(H) be an (α, β)-normal operator and λ ∈ ℂ. Then

(2.7)  α∥T ∥^{2} ≤ w(T^{2}) + \frac{2β∥T − λT^{∗}∥^{2}}{(1 + |λ|α)^{2}} .

Proof. Using the Dunkl–Williams inequality [8]

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

we get

2 − 2 · \frac{Re⟨a, b⟩}{∥a∥∥b∥} = ∥\frac{a}{∥a∥} − \frac{b}{∥b∥}∥^{2} ≤ \frac{4∥a − b∥^{2}}{(∥a|| + ∥b∥)^{2}} (a, b ∈ H \ \{0\})

whence

∥a∥∥b∥ ≤ \frac{2∥a∥ ∥b∥ ∥a − b∥^{2}}{(∥a|| + ∥b∥)^{2}} + |⟨a, b⟩| (a, b ∈ H \ \{0\}).

Put a = T x and b = λT^{∗} to get

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

so that

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

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

Taking the supremum in (2.8) over x ∈ H, ∥x∥ = 1, we get the desired result (2.7).

Theorem 2.4. Let T ∈ B(H) be an (α, β)-normal operator and λ ∈ ℂ\{0}. Then

(2.9)  [α^{2} − (\frac{1}{|λ|} + β)^{2}] ∥T ∥^{4} ≤ w(T^{2}).

Proof. We apply the following reverse of the quadratic Schwarz inequality obtained by Dragomir in [5]

(2.10)  (0 ≤)∥a∥^{2}∥b∥^{2} − |⟨a, b⟩|^{2} ≤ \frac{1}{|λ|^{2}} ∥a∥^{2}∥a − λb∥^{2}

provided a, b ∈ H and λ ∈ ℂ\{0}.

Set a = T x, b = T^{∗}x in (2.10), to get

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

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

whence

(2.11)  [α^{2} − (\frac{1}{|λ|} + β)^{2}] ∥T x∥^{4} ≤ |⟨T^{2}x, x⟩|^{2}.

Taking the supremum in (2.11) over x ∈ H, ∥x∥ = 1, we get the desired result (2.9).

Theorem 2.5. Let T ∈ B(H) be an (α, β)-normal operator, r ≥ 0 and λ ∈ ℂ⧹ {0}. If ∥λT^{∗} − T ∥ ≤ r and \frac{r}{|λ|} ≤ inf\{∥T^{∗}x∥ : ∥x∥ = 1\}, then

(2.12)  α^{2}∥T ∥^{4} ≤ w(T^{2})^{2} + \frac{r^{2}}{|λ|^{2}} ∥T ∥^{2}.

Proof. We use the following reverse of the Schwarz inequality obtained by Dragomir in [3] (see also [4, p. 20]):

(2.13)  (0 ≤) ∥y∥^{2} ∥a∥^{2} − [Re⟨y, a⟩]^{2} ≤ r^{2}∥y∥^{2},

provided ∥y − a∥ ≤ r ≤ ∥a∥.

By the assumption of theorem ∥T x − λT^{∗}x∥ ≤ r ≤ ∥λT^{∗}x∥. Setting a = λT^{∗}x and y = T x, with ∥x∥ = 1 in (2.13) we get

∥T x∥^{2} ∥λT^{∗}x∥^{2} ≤ [Re⟨T x, λT^{∗}x⟩]^{2} + r^{2}∥T x∥^{2}

whence

(2.14)  α^{2}|λ|^{2}∥T x∥^{4} ≤ |λ|^{2}|⟨T^{2}x, x⟩|^{2} + r^{2}∥T x∥^{2}.

Taking the supremum in (2.14) over x ∈ H, ∥x∥ = 1, we get the desired result (2.12).

Finally, the following result that is less restrictive for the involved parameters r and λ (from the above theorem) may be stated as well:

Theorem 2.6. Let T ∈ B(H) be an (α, β)-normal operator, r ≥ 0 and λ ∈ ℂ⧹ {0}. If ∥λT^{∗} − T ∥ ≤ r, then

(2.15)  α∥T ∥^{2} ≤ w(T^{2}) + \frac{r^{2}}{2|λ|}.

Proof. We use the following reverse of the Schwarz inequality obtained by Dragomir in [2] (see also [4, p. 27]):

(2.16)  (0 ≤) ∥y∥ ∥a∥ − Re⟨y, a⟩ ≤ \frac{1}{2}r^{2},

provided ∥y − a∥ ≤ r.

Setting a = λT^{∗}x and y = T x, with ∥x∥ = 1 in (2.16) we get

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

which gives

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

Now, taking the supremum over ∥x∥ = 1 in this inequality, we get the desired result (2.15)

3. Inequalities Involving Norms

Our first result in this section reads as follows.

Theorem 3.1. Let T ∈ B(H) be an (α, β)-normal operator. If p ≥ 2, then

(3.1)  2(1 + α^{p})∥T ∥^{p} ≤ \frac{1}{2}(∥T + T^{∗}∥^{p} + ∥T − T^{∗}∥^{p}).

In general, for each T ∈ B(H) and p ≥ 2 we have

(3.2)  ∥\frac{T^{∗}T + T T^{∗}}{2} ∥^{p/2} ≤ \frac{1}{4}(∥T + T^{∗}∥^{p} + ∥T − T^{∗}∥^{p}).

Proof. We use the following inequality obtained by Dragomir and Sándor in [7] (see also [11, p. 544]):

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

for any a, b ∈ H and p ≥ 2.

Now, if we choose a = T x, b = T^{∗}x in (3.3), then we get

(3.4)  ∥T x + T^{∗}x∥^{p} + ∥T x − T^{∗}x∥^{p} ≥ 2(∥T x∥^{p} + ∥T^{∗}x∥^{p}),

whence

(3.5)  ∥T x + T^{∗}x∥^{p} + ∥T x − T^{∗}x∥^{p} ≥ 2(∥T x∥^{p} + α^{p}∥T x∥^{p}),

for any x ∈ H, ∥x∥ = 1.

Taking the supremum in (3.5) over x ∈ H, ∥x∥ = 1, we get the desired result (3.1).

Now for the general case T ∈ B(H), observe that

(3.6)  ∥T x∥^{p} + ∥T^{∗}x∥^{p} = (∥T x∥^{2})^{\frac{p}{2}} + (∥T^{∗}x∥^{2})^{\frac{p}{2}}

and by applying the elementary inequality:

\frac{a^{q} + b^{q}}{2} ≥ (\frac{a + b}{2})^{q} , a, b ≥ 0 and q ≥ 1

we have

(3.7) (∥T x∥^{2})^{\frac{p}{2}} + (∥T^{∗}x∥^{2})^{\frac{p}{2}} ≥ 2^{1− \frac{p}{2}} (∥T x∥^{2} + ∥T^{∗}x∥^{2})^{\frac{p}{2}}

= 2^{1− \frac{p}{2}} [⟨T x, T x⟩ + ⟨T^{∗}x, T^{∗}x⟩]^{\frac{p}{2}}

= 2^{1− \frac{p}{2}} [⟨(T^{∗}T + T T^{∗})x, x⟩]^{\frac{p}{2}} .

Combining (3.4) with (3.7) and (3.6) we get

(3.8)  \frac{1}{4}[∥T x − T^{∗}x∥^{p} + ∥T x + T^{∗}x∥^{p}] ≥ |⟨(\frac{T^{∗}T + T T^{∗}}{2} ) x, x⟩|^{p/2}

for any x ∈ H, ∥x∥ = 1. Taking the supremum over x ∈ H, ∥x∥ = 1, and taking into account that

w (\frac{T^{∗}T + T T^{∗}}{2} ) = ∥\frac{T^{∗}T + T T^{∗}}{2} ∥ ,

we deduce the desired result (3.2).

Theorem 3.2. Let T ∈ B(H) be an (α, β)-normal operator. If p ∈ (1, 2) and λ, μ ∈ ℂ, then

(3.9) [(|λ| + β|μ|)^{p} + max{|λ| − |μ|β, α|μ| − |λ|}] ∥T ∥^{p} ≤ ∥λT + μT^{∗}∥^{p} + ∥λT − μT^{∗}∥^{p}.

Proof. We use the following inequality obtained by Dragomir and Sándor in [7] (see also [11, p. 544])

(3.10)  (∥a∥ + ∥b∥)^{p} + | ∥a∥ − ∥b∥ |^{p} ≤ ||a + b∥^{p} + ∥a − b∥^{p},

for any a, b ∈ H and p ∈ (1, 2).

Put a = λT x, b = μT^{∗}x in (3.10) to obtain

(∥λT x∥ + ∥μT^{∗}x∥)^{p} + |∥λT x∥ − ∥μT^{∗}x∥|^{p}

≤ ∥λT x + μT^{∗}x∥^{p} + ∥λT x − μT^{∗}x∥^{p},

whence

(3.11) (|λ| + |μ|α)^{p}∥T x∥^{p} + (max{|λ| − |μ|β, α|μ| − |λ|}) ∥T x∥^{p} ≤ ∥λT x + μT^{∗}x∥^{p} + ∥λT x − μT^{∗}x∥^{p},

for any x ∈ H, ∥x∥ = 1.

Taking the supremum in (3.11) over x ∈ H, ∥x∥ = 1, we get the desired result (3.9).

4. Other Inequalities for General Operators

Finally, we present two results holding in the general case of bounded linear operators in Hilbert spaces:

Theorem 4.1. Let T, S ∈ B(H). Then

(4.1)  | ∥T^{∗}T + S^{∗}S∥ − ∥T + S∥^{2}| ≤ 2w(S^{∗}T),

in particular,

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

Proof. We have

(4.3)  ∥T x ± Sx∥^{2} = ∥T x∥^{2} ± 2Re⟨T x, Sx⟩ + ∥Sx∥^{2}

for any x ∈ H. Hence

∥T x + Sx∥^{2} ≤ ⟨(T^{∗}T + S^{∗}S)x, x⟩ + 2|⟨(S^{∗}T)x, x⟩|.

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

∥T + S∥^{2} ≤ w(T^{∗}T + S^{∗}S) + 2w(S^{∗}T) = ∥T^{∗}T + S^{∗}S∥ + 2w(S^{∗}T),

It follows from (4.3) that

⟨(T^{∗}T + S^{∗}S)x, x⟩ = ∥T x∥^{2} + ∥Sx∥^{2}

= 2Re⟨T x, Sx⟩ + ∥T x − Sx∥^{2}

≤ 2|⟨T x, Sx⟩| + ∥T x − Sx∥^{2}.

Replacing S by −S in the later equality and taking the supremum over x ∈ H, ∥x∥ = 1, we get

∥T^{∗}T + S^{∗}S∥ = w(T^{∗}T + S^{∗}S) ≤ 2w(S^{∗}T) + ∥T + S∥^{2} .

The desired inequality (4.1) follows from (1.2) and (1.3). The last inequality can be obtained by putting S = λT^{∗} in (4.1).

Theorem 4.2. Let T, S ∈ B(H), and p, q > 0. Then

(4.4)  w(T + S) ≤ [w (\frac{p + q}{p} · T^{∗}T + \frac{p + q}{q} · S^{∗}S)]^{1/2} .

In particular,

w(T + λT^{∗}) ≤ [w (\frac{p + q}{p} · T^{∗}T + \frac{p + q}{q} |λ|^{2}T T^{∗})]^{1/2} .

Proof. Utilizing the following elementary inequality

\frac{(a + b)^{2}}{p + q} ≤ \frac{a^{2}}{p} + \frac{b^{2}}{q} ,

holding for any real numbers a, b and for the positive numbers p, q, we get

|⟨(T + S)x, x⟩|^{2} ≤ (p + q)\frac{(|⟨T x, x⟩| + |⟨Sx, x⟩|)^{2}}{p + q}

≤ (p + q) (\frac{|⟨T x, x⟩|^{2}}{p} + \frac{|⟨Sx, x⟩|^{2}}{q} )

≤ (p + q) (\frac{∥T x∥^{2}}{p} + \frac{∥Sx∥^{2}}{q} )

≤ (p + q) (⟨\frac{T^{∗}T}{p} x, x⟩ + ⟨\frac{S^{∗}S}{q} x, x⟩)

≤ ⟨(\frac{p + q}{p} T^{∗}T + \frac{p + q}{q} S^{∗}S) x, x⟩ .

Putting S = λT^{∗} in (4.4), we get the last desired inequality.

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1 School of Computer Science and Mathematics, Victoria University, P. O. Box 14428, Melbourne City, Victoria 8001, Australia.

E-mail address: sever.dragomir@vu.edu.au

2 Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran; Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University, Iran.

E-mail address: moslehian@ferdowsi.um.ac.ir and moslehian@ams.org