A generalized Hölder-type inequalities for measurable operators

We prove a generalized Hölder-type inequality for measurable operators associated with a semi-finite von Neumann algebra which is a generalization of the result shown by Bekjan (Positivity 21:113–126, 2017). This also provides a generalization of the unitarily invariant norm inequalities for matrix due to Bhatia–Kittaneh, Horn–Mathisa, Horn–Zhan and Zou under a cohyponormal condition.


Introduction
Let M n be the space of n × n complex matrices. A norm ||| · ||| on M n is called unitarily invariant if |||UAV ||| = |||A||| for all A ∈ M n and all unitary matrices U, V ∈ M n . Let A, B ∈ M n . In 1990, Bhatia and Kittaneh [6] established an arithmetic-geometric mean inequality for unitarily invariant norms, i.e., (1.1) Using tensor algebra techniques, a strengthening inequality of (1.1) was presented by Bhatia and Davis [5] which is the Hölder inequality for unitarily invariant norms. In particular, these authors also showed in [16] that Subsequently, a considerable different proofs, equivalent statements, along with some generalizations, refinements, and applications of inequalities (1.1)-(1.4) were discussed by many authors. We refer to [1-3, 5, 15, 20] for more information on this topic and historical references. Let A, B ∈ M n and 1 p + 1 q = 1, p, q > 1, α ∈ [0, 1], r ≥ 0 and let T X (α) = αAA * X + (1α)XBB * . In 2015, by majorization techniques, Audenaert [2] prove an inequality that interpolates between the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities Recently, Zou [20] presented the inequality for unitarily invariant norms which is a unified version of inequalities (1.1) and (1.6). By the concept of uniform Hardy-Littlewood majorization Bekjan [8] gave a Höldertype inequality (1.4) for τ -measurable operators associated with a semi-finite von Neumann algebra and for symmetric Banach spaces norm. In this paper, we will give a generalized Hölder-type inequality (1.7) for τ -measurable operators under a cohyponormal condition by adopting a technique similar to the one used by Bekjan and Zou. This is a generalization of Bekjan's result in [8].

Preliminaries
Let L 0 be the set of all Lebesgue measurable functions on (0, ∞). A Banach space E ⊆ L 0 with the norm · E satisfying the condition that f ∈ E and f E ≤ g E whenever 0 ≤ f ≤ g, f ∈ L 0 and g ∈ F, is said to be a Banach function space and m denotes the Lebesgue measure on (0, ∞). The symmetric Banach function space E is called fully if and only if f ∈ E, g ∈ L 0 and t 0 f * (s) ds ≥ t 0 g * (s) ds give us that g ∈ E and f E ≥ g E . We say that E has order continuous norm if for every net {f i } i∈Λ ⊆ E such that f i ↓ 0 we have f i E ↓ 0. In particular, a symmetric Banach function space which has order continuous norm is automatically fully symmetric. For 0 < r < ∞, E (r) will denote the quasi-Banach spaces defined by E (r) := g ∈ L 0 : |g| r ∈ E and g E (r) = |g| r 1 r E .
For r > 0, we know from [17] that if E is a symmetric Banach function space, then E (r) is a symmetric quasi-Banach space, and if E has order continuous norm, then E (r) has order continuous norm.
We suppose that M is a semi-finite von Neumann algebra, namely a von Neumann algebra equipped with a semi-finite, faithful and normal trace τ . We will denote by 1 the identity in M and P(M) the projection lattice of M. A closed densely defined linear operator x in H with domain D(x) ⊆ H is said to be affiliated with M if u * xu = x for all unitary operators u which belong to the commutant M of M. Let e ⊥ s (|x|) = e (s,∞) (|x|) be the spectral projection of |x| associated with the interval (s, ∞). If x is affiliated with M, x will be called τ -measurable if and only if τ (e ⊥ s (|x|)) < ∞ for some s > 0. The set of all τ -measurable operators will be denoted by L 0 (M).
We will denote simply by λ(x) and μ(x) the functions t → λ t (x) and t → μ t (x), respectively. The generalized singular number function t → μ t (x) is decreasing rightcontinuous. For x, y ∈ L 0 (M) and u, v ∈ M, we obtain Moreover, let f be a continuous increasing function on [0, ∞) with f (0) = 0. It follows from [11, Lemma 2.5, Lemma 2.6 and Corollary 2.8] that See [11] for basic properties and detailed information on generalized singular number of x. Let E be a symmetric Banach function space on (0, ∞). We define As is shown in [10, Proposition 3.1], if E is a symmetric Banach function space, then is also a noncommutative fully symmetric Banach function space when r ≥ 1 and E is fully (cf. [19]). In the following, unless stated otherwise, we will keep all previous notations throughout the paper, and we always assume that E is a symmetric Banach function space on (0, ∞) with order continuous norm.

Main results
We start this section with several lemmas which will be used in our proof. From [ Recall that an operator x ∈ L 0 (M) is said to be hyponormal if x * x ≥ xx * , cohyponormal if x * is hyponormal. Since xx * (yy * ) α is cohyponormal, [ This completes the proof.
Remark 3.8 It is necessary for us to remark here that, it can be observed in [7,Lemma 2] without a proof that μ(ab) = μ(ba) when ab, ba ∈ L 1 (M). However, we are not able to give it a proof at this moment. On the other hand, the authors were informed by an anonymous referee that μ(ab) = μ(ba) does not hold even in the matrix case. On account of this, there could be a gap in the proof of [13,Theorem 3.6] and we give a corresponding illustration as follows: Set r ≥ 1, α ∈ [0, 1] and let xx * (yy * ) α be cohyponormal. Using Proposition 3.6 to the case E = L 1 and p = q = 2, we have i.e., which is the result of [14,Theorem 3.6] under a cohyponormal condition.