More results on generalized singular number inequalities of τ-measurable operators

Let \(\mathbb{M}_{n}\) be the space of \(n\times n\) complex matrices. Given \(A\in\mathbb{M}_{n}\), we define \(|A|=(A^{*}A)^{\frac{1}{2}}\). The singular values of A, i.e., the eigenvalues of the operator \(|A|\), enumerated in decreasing order, will be denoted by \(S_{j}(A)\), \(j=1, 2, \ldots, n\). The arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh [1] says that

$$ 2S_{j} \bigl(AB^{*} \bigr)\leq S_{j} \bigl(A^{*}A+B^{*}B \bigr),\quad j=1, 2, \ldots, n, $$

(1.1)

holds for any \(A, B\in\mathbb{M}_{n}\). In 2000, Zhan [2] proved that

$$ 2S_{j}(A-B)\leq S_{j}(A\oplus B), \quad j=1, 2, \ldots, n, $$

(1.2)

for positive semidefinite matrices \(A, B\in\mathbb{M}_{n}\). On the other hand, Tao [3] observed that if \(A, B, K\in\mathbb{M}_{n}\) with \(\bigl ( {\scriptsize\begin{matrix}{} A&K\cr K^{*}&B \end{matrix}} \bigr )\geq0\), then

$$ 2S_{j}(K)\leq S_{j} \left ( \begin{pmatrix} A&K\\ K^{*}&B \end{pmatrix} \right ),\quad j=1, 2, \ldots, n. $$

(1.3)

It was pointed out in [3] that inequalities (1.1), (1.2), and (1.3) are equivalent. According to inequality (1.3), Audenaert [4] (see also [5]) gave a Heinz mean inequality for singular values, that is, if \(A, B\in\mathbb{M}_{n}\) are positive semidefinite matrices and \(0\leq r\leq1\), then

$$ S_{j} \bigl(A^{r}B^{1-r}+A^{1-r}B^{r} \bigr)\leq S_{j}(A+B),\quad j=1, 2, \ldots, n. $$

(1.4)

Among other things, in 2012, Albadawi [6] showed that if \(A_{i}, B_{i}, X_{i}\in B(\mathcal{H})\) (\(i=1, 2,\ldots, n\)) with \(X_{i}\geq0\), then

(1.5)

holds for \(j=1, 2, \ldots\) . Inequality (1.5) yields the well-known arithmetic-geometric mean inequality for singular values as special cases.

Using the notion of the generalized singular number studied by Fack and Kosaki [7], we generalize inequalities (1.1)-(1.5) for τ-measurable operators associated with a semifinite von Neumann algebra \(\mathcal{M}\).

Unless stated otherwise, \(\mathcal{M}\) will always denote a semifinite von Neumann algebra acting on a Hilbert space \(\mathcal{H}\), with a normal faithful semifinite trace τ. We refer to [7, 8] for noncommutative integration. We denote the identity of \(\mathcal{M}\) by 1 and let \(\mathcal{P}\) denote the projection lattice of \(\mathcal{M}\). A closed densely defined linear operator x in \(\mathcal{H}\) with domain \(D(x)\subseteq\mathcal{H}\) is said to be affiliated with \(\mathcal{M}\) if \(u^{*}xu=x\) for all unitary operators u which belong to the commutant \(\mathcal{M^{\prime}}\) of \(\mathcal {M}\). If x is affiliated with \(\mathcal{M}\), we define its distribution function by \(\lambda_{s}(x)=\tau(e^{\bot}_{s}(|x|))\) and x will be called τ-measurable if and only if \(\lambda_{s}(x)<\infty \) for some \(s>0\), where \(e^{\bot}_{s}(|x|)=e_{(s, \infty)}(|x|)\) is the spectral projection of \(|x|\) associated with the interval \((s, \infty)\). The set of all τ-measurable operators will be denoted by \(\overline{ \mathcal{M}} \). The set \(\overline{\mathcal{M}} \) is a ∗-algebra with sum and product being the respective closures of the algebraic sum and product.

Let \(\mathbb{M}_{n}(\mathcal{M})\) denote the linear space of \(n\times n\) matrices

with entries \(x_{ij}\in\mathcal{M}\), \(i,j=1,2,\ldots,n\). Let \(\mathcal{H}^{n}=\bigoplus_{i=1}^{n}\mathcal{H}\). Then \(\mathbb{M}_{n}(\mathcal{M})\) is a von Neumann algebra in the Hilbert space \(\mathcal{H}^{n}\). For \(x\in\mathbb{M}_{n}(\mathcal {M})\), define \(\tau_{n}(x)=\sum_{i=1}^{n}\tau(x_{ii})\), then \(\tau_{n}\) is a normal faithful semifinite trace on \(\mathbb{M}_{n}(\mathcal{M})\). The direct sum of operators \(x_{1}, x_{2},\ldots, x_{n}\in \overline{\mathcal {M}}\), denoted by \(\bigoplus_{i=1}^{n}x_{i}\), is the block-diagonal operator matrix defined on \(\mathcal{H}^{n}\) by

The following lemma, which includes a basic property of generalized singular number, plays a central role in our investigation.

As an application of Lemma 3.1 we now obtain the desired generalized singular number inequality (1.3) for τ-measurable operators.

Combing Lemma 3.3 with the following theorem we see that inequalities (1.1), (1.2), and (1.3) hold for τ-measurable operators.

Based on Lemma 3.5 we now obtain the desired generalized singular number inequality (1.4) for τ-measurable operators.

We conclude this section with a series of inequalities which are related to the Heinz mean inequality for a generalized singular number of τ-measurable operators.

In this section, we establish a generalized singular number inequality for τ-measurable operators which yields the well-known arithmetic-geometric mean inequalities as special cases.

The following proposition is a refinement of the inequality in Theorem 3.4(2).

From Proposition 4.1 we now obtain the promised generalized singular number inequality (1.5) for τ-measurable operators.

Proposition 4.2

Let \(x_{i}, y_{i}\in \overline{\mathcal{M}} \) and \(0\leq z_{i}\in\mathcal{M}\) (\(i=1, 2, \ldots, n\)). Then

Proof

Let

Then

From Proposition 4.1, we have

$$2\mu_{t} \bigl(AKB^{*} \bigr)\leq\|K\|\mu_{t} \bigl(A^{*}A+B^{*}B \bigr)= \|K\|\mu_{t} \bigl(|T|^{2} \bigr)=\|K\| \mu_{t}(T)^{2}, \quad t>0. $$

Then the result follows from Lemma 3.1. □

Proposition 4.2 includes several generalized singular number inequalities as special cases.

The following inequality is an application of Corollary 4.3.

The following inequality contains a generalization of the inequality in Theorem 3.4(1).