Skew-Hermitian operators in real Banach spaces of self-adjoint compact operators

TL;DR

This paper characterizes skew-Hermitian operators on real Banach subspaces of self-adjoint compact operators in infinite-dimensional Hilbert spaces, showing they are generated by bounded self-adjoint operators via a specific commutator form.

Contribution

It provides a new representation theorem for skew-Hermitian operators in certain Banach ideals of compact operators, extending understanding of their structure in infinite-dimensional settings.

Findings

Skew-Hermitian operators are of the form $i(xa - ax)$ for some bounded self-adjoint $a$.

The result holds for separable or perfect Banach symmetric ideals, excluding the Hilbert-Schmidt class.

The theorem generalizes known finite-dimensional results to infinite-dimensional Banach operator ideals.

Abstract

Let $\mathcal H$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal K(\mathcal H)$ be the $C^*$-algebra of compact linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a symmetric sequence space. If $\{\mu(n,x)\}$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{\mu(n,x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{\mu(n,x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h=\{x\in\mathcal C_E : x=x^*\}$ be the real Banach subspace of self-adjoint operators in $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$. We show that in the case when $\mathcal C_E$ is a separable or perfect Banach symmetric ideal, $\mathcal C_E \neq \mathcal C_{l_2}$, for any skew-Hermitian operator $H\colon\mathcal C_E^h \to \mathcal C_E^h$ there exists self-adjoint bounded linear operator $a$ in $\mathcal H$ such that $H(x)=i(xa - ax)$ for all $x\in\mathcal C_E^h$.

Equations

$$\mathcal{C}_{p}:=\Big{\{}x\in\mathcal{K}(\mathcal{H})\,:\,\mathrm{Tr}\big{(}|x|^{p}\big{)}<\infty\Big{\}}\ \ (respectively,\mathcal{C}_{\infty}=\mathcal{K}(\mathcal{H}))$$

$$\|x\|_{p}:=\mathrm{Tr}\big{(}|x|^{p}\big{)}^{\frac{1}{p}},\quad(respectively,\ \|x\|_{\infty}:=\sup\limits_{n\geq 1}|\mu(n,x)|).$$

$${\mathcal{L}}_1(\mathbb{N},2^{\mathbb{N}},\mu )={\ell }_1=\left\{\{{\xi }_n{\}}_{n=1}^{\infty }\subset \mathbb{C}:\Vert \{{\xi }_n\}{\Vert }_1=\sum _{n=1}^{\infty }|{\xi }_n|<\infty \right\}\subset {\ell }_{\infty },$$

$${\xi }^{\ast }(t)=\inf \{\lambda :\mu \{|\xi |>\lambda \}\le t\},t>0,$$

$${\xi }_n^{\ast }=\inf \left\{\underset{n\notin F}{\sup }|{\xi }_n|:F\subset \mathbb{N},|F|<n\right\}.$$

$$\xi =\{{\xi }_n\}\prec \prec \eta =\{{\eta }_n\}⟺\sum _{n=1}^m{\xi }_n^{\ast }\le \sum _{n=1}^m{\eta }_n^{\ast }\text{ for all }m\in \mathbb{N}.$$

$$\eta \in E,\xi \in {\ell }_{\infty },{\xi }^{\ast }\le {\eta }^{\ast }\text{ (resp.},{\xi }^{\ast }\prec \prec {\eta }^{\ast })⟹\xi \in E\text{ and }\Vert \xi {\Vert }_E\le \Vert \eta {\Vert }_E.$$

$$\{{\xi }_n{\}}_{n=1}^{\infty }=\xi \le \eta =\{{\eta }_n{\}}_{n=1}^{\infty }⟺{\xi }_n\le {\eta }_n\text{for all}n\in \mathbb{N},$$

$$\Vert \xi {\Vert }_E=\Vert |\xi |{\Vert }_E=\Vert {\xi }^{\ast }{\Vert }_E\text{ for all }\xi \in E_h.$$

$${\ell }_p=\left\{\xi =\{{\xi }_n{\}}_{n=1}^{\infty }\in c_0:\Vert \xi {\Vert }_p={\left(\sum _{n=1}^{\infty }|{\xi }_n{|}^p\right)}^{1/p}<\infty \right\},1\le p<\infty .$$

$$({\ell }_1,\Vert \cdot {\Vert }_1)\subset (E,\Vert \cdot {\Vert }_E)\subset ({\ell }_{\infty },\Vert \cdot {\Vert }_{\infty }).$$

$$\mathrm{Tr}(x)=\sum _{j\in J}(x{\phi }_j,{\phi }_j),x\in {\mathcal{B}}_{+}(\mathcal{H}),$$

$${\mu }_t(x)=\inf \{\lambda >0:\mathrm{Tr}(e_{\lambda }^{\perp })\le t\}$$

$$x\in X,\ y\in\mathcal{B}(\mathcal{H}),\ \mu_{t}(y)\leq\mu_{t}(x)\text{ \ \ for all\ \ }t>0\$$

$$x\in X,y\in \mathcal{B}(\mathcal{H}),\underset{0}{\overset{s}{\int }}{\mu }_t(y)dt\le \underset{0}{\overset{s}{\int }}{\mu }_t(x)dt\text{ for all }s>0(\text{writing}y\prec \prec x))$$

$${\mathcal{C}}_p=\{x\in \mathcal{K}(\mathcal{H}):\Vert x{\Vert }_p=\mathrm{Tr}(|x{|}^p{)}^{1/p}<\infty \},1\le p<\infty ,$$

$$\Vert x{\Vert }_X=\Vert |x|{\Vert }_X=\Vert x^{\ast }{\Vert }_X,axb\in X,\text{ and }\Vert axb{\Vert }_X\le \Vert a{\Vert }_{\infty }\Vert b{\Vert }_{\infty }\Vert x{\Vert }_X.$$

$$E(X)=\left\{\xi =\{{\xi }_n{\}}_{n=1}^{\infty }\in c_0:x_{\xi }=\sum _{n=1}^{\infty }{\xi }_n{p}_n\in X\right\}$$

$${\mathcal{C}}_E=\{x\in \mathcal{K}(\mathcal{H}):\{\mu (n,x)\}\in E\},\Vert x{\Vert }_{{\mathcal{C}}_E}=\Vert \{\mu (n,x)\}{\Vert }_E.$$

$$E({\mathcal{C}}_E)=E,\Vert \cdot {\Vert }_{E({\mathcal{C}}_E)}=\Vert \cdot {\Vert }_E,{\mathcal{C}}_{E({\mathcal{C}}_E)}={\mathcal{C}}_E,\Vert \cdot {\Vert }_{{\mathcal{C}}_{E({\mathcal{C}}_E)}}=\Vert \cdot {\Vert }_{{\mathcal{C}}_E}.$$

$$E^{\times }=\{\xi =\{{\xi }_n{\}}_{n=1}^{\infty }\in {\ell }_{\infty }:\xi \eta =\{{\xi }_n{\eta }_n{\}}_{n=1}^{\infty }\in {\ell }_1\text{for all}\eta =\{{\eta }_n{\}}_{n=1}^{\infty }\in E\},$$

$$(\text{respectively},{\mathcal{C}}_E^{\times }=\{x\in \mathcal{B}(\mathcal{H}):xy\in {\mathcal{C}}_1\text{for all}y\in {\mathcal{C}}_E\}),$$

$$\Vert \xi {\Vert }_{E^{\times }}=\sup \{\sum _{n=1}^{\infty }|{\xi }_n{\eta }_n|:\eta =\{{\eta }_n{\}}_{n=1}^{\infty }\in E,\Vert \eta {\Vert }_E\le 1\},\xi \in E^{\times },$$

$$(\text{respectively},\ \ \|x\|_{\mathcal{C}_{E}^{\times}}=\sup\limits\{\mathrm{Tr}\big{(}|xy|\big{)}:y\in\mathcal{C}_{E},\ \|y\|_{\mathcal{C}_{E}}\leq 1\},\ x\in\mathcal{C}_{E}^{\times}).$$

$$0\le {\xi }_k\le {\xi }_{k+1},{\xi }_k\in E\text{(respectively},0\le x_k\le x_{k+1},x_k\in {\mathcal{C}}_E)\text{for all}k\in \mathbb{N}$$

$$⟨x,\xi \otimes \eta ⟩:=\mathrm{Tr}((\eta \otimes \xi )\cdot x)=(x(\eta ),\xi )$$

$$f_y(x):=⟨x,y⟩=\mathrm{Tr}(y^{\ast }x),x\in {\mathcal{C}}_E^h,$$

$$f_y(y)=\mathrm{Tr}(y^2)=\mathrm{Tr}(y)=(\xi ,\xi )=1=\Vert y{\Vert }_{{\mathcal{C}}_E}^2.$$

$$|f_y(x)|=|\mathrm{Tr}(yx)|=|(x(\xi ),\xi )|\le \Vert x{\Vert }_{\infty }\le \Vert x{\Vert }_{{\mathcal{C}}_E}\le 1.$$

$$0=[H(p),p]=f_p(H(p))=⟨H(p),p⟩.$$

Full text

1. Introduction

Let $(\mathcal{H},(\cdot,\cdot))$ be an infinite-dimensional complex separable Hilbert space, and let $\mathcal{B}(\mathcal{H})$ (respectively, $\mathcal{K}(\mathcal{H})$) be the $C^{*}$-algebra of all bounded (respectively, compact) linear operators on $\mathcal{H}$. For a compact operator $x\in\mathcal{K}(\mathcal{H})$, we denote by \big{\{}\mu(n,x)\big{\}}_{n=1}^{\infty} the singular value sequence of $x$,  that is, the decreasing rearrangement of the eigenvalue sequence of $|x|=(x^{*}x)^{\frac{1}{2}}$. We let $\mathrm{Tr}$ denote the standard trace on $\mathcal{B}(\mathcal{H})$. For $p\in[1,\infty)$ ($p=\infty$), we let

[TABLE]

denote the $p$-th Schatten ideal of $\mathcal{B}(\mathcal{H})$, with the norm

[TABLE]

The Schatten ideals $\mathcal{C}_{p}=\mathcal{C}_{l_{p}}$ are examples of Banach symmetric ideals \mathcal{C}_{E}=\{x\in\mathcal{K}(\mathcal{H}):\big{\{}\mu(n,x)\big{\}}_{n=1}^{\infty}\in E\} with norm \|x\|_{\mathcal{C}_{E}}=\|\big{\{}\mu(n,x)\big{\}}_{n=1}^{\infty}\|_{E} of compact operators generated by symmetric sequence spaces $(E,\|\cdot\|_{E})$ (see section 2 below).

Let $[\cdot,\cdot]$ be a semi-inner product on $\mathcal{C}_{E}$ compatible with the norm $\|\cdot\|_{\mathcal{C}_{E}}$, that is, $\|x\|_{\mathcal{C}_{E}}=\sqrt{[x,x]}$ for all $x\in\mathcal{C}_{E}$  [6, Ch. 2, §1]. A bounded linear operator $H:\mathcal{C}_{E}\to\mathcal{C}_{E}$ is called Hermitian if $[Hx,x]$ is real for all $x\in\mathcal{C}_{E}$ [9, Ch. 5, §2].

In 1981 A.Sourour [17] gave the following description of all the Hermitian operators acting in separable Banach symmetric ideal.

Theorem 1.1**.**

Let $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ be a separable Banach symmetric ideal, and let $\mathcal{C}_{E}\neq\mathcal{C}_{2}$. Then for any Hermitian operator $H\colon\mathcal{C}_{E}\to\mathcal{C}_{E}$   there are self-adjoint operators $a,b\in\mathcal{B}(\mathcal{H})$ such that $H(x)=ax+xb$ for all $x\in\mathcal{C}_{E}$.

In [2], a variant of Theorem 1.1 was obtained for any perfect Banach symmetric ideals $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}}),\ \mathcal{C}_{E}\neq\mathcal{C}_{2}$  ( recall that $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ is a perfect ideals, if $\mathcal{C}_{E}=\mathcal{C}_{E}^{\times\times}$ [11] (see section 2 below)).

Let $\mathcal{C}_{E}^{h}=\{x\in\mathcal{C}_{E}:x=x^{*}\}$ be a Banach real subspace in Banach symmetric ideals $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$. A linear bounded operator $H\colon\mathcal{C}_{E}^{h}\to\mathcal{C}_{E}^{h}$ is said to be skew-Hermitian, if $[H(x),x]=0$ for all $x\in\mathcal{C}_{E}^{h}$, where $[\cdot,\cdot]$ is a semi-inner product on $\mathcal{C}_{E}$ compatible with the norm $\|\cdot\|_{\mathcal{C}_{E}}$.

It is clear that the linear operator $H\colon\mathcal{C}_{E}^{h}\to\mathcal{C}_{E}^{h}$ defined by $H(x)=i(xa-ax)$, where $a=a^{*}\in\mathcal{B}(\mathcal{H})$,  $i^{2}=-1$,  is a skew-Hermitian operator.

Our main result states that if $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ is a separable or a perfect Banach symmetric ideal of compact operators, $\mathcal{C}_{E}\neq\mathcal{C}_{2}$, then there are no other skew-Hermitian operators in $(\mathcal{C}_{E}^{h},\|\cdot\|_{\mathcal{C}_{E}})$:

Theorem 1.2**.**

Let $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ be a separable or a perfect Banach symmetric ideal,   $\mathcal{C}_{E}\neq\mathcal{C}_{2}$, and let $H\colon\mathcal{C}_{E}^{h}\to\mathcal{C}_{E}^{h}$ be a skew-Hermitian operator. Then there exists self-adjoint operator $a\in\mathcal{B}(\mathcal{H})$ such that $H(x)=i(xa-ax)$ for all $x\in\mathcal{C}_{E}^{h}$.

2. Preliminaries

Let $\ell_{\infty}$ (respectively, $c_{0}$) be the Banach space of bounded (respectively, converging to zero) sequences $\{\xi_{n}\}_{n=1}^{\infty}$ of complex numbers equipped with the norm $\|\{\xi_{n}\}\|_{\infty}=\sup\limits_{n\in\mathbb{N}}|\xi_{n}|$, where $\mathbb{N}$ is the set of natural numbers. If $2^{\mathbb{N}}$ is the $\sigma$-algebra of subsets of $\mathbb{N}$ and $\mu(\{n\})=1$ for each $n\in\mathbb{N}$, then $(\mathbb{N},2^{\mathbb{N}},\mu)$ is a $\sigma$-finite measure space such that $\mathcal{L}_{\infty}(\mathbb{N},2^{\mathbb{N}},\mu)=\ell_{\infty}$ and

[TABLE]

where $\mathbb{C}$ is the field of complex numbers.

If $\xi=\{\xi_{n}\}_{n=1}^{\infty}\in\ell_{\infty}$, then the non-increasing rearrangement $\xi^{*}:(0,\infty)\to(0,\infty)$ of $\xi$ is defined by

[TABLE]

(see, for example, [4, Ch. 2, Definition 1.5]). As such, the non-increasing rearrangement of a sequence $\{\xi_{n}\}_{n=1}^{\infty}\in\ell_{\infty}$ can be identified with the sequence $\xi^{*}=\{\xi_{n}^{*}\}_{n=1}^{\infty}$, where

[TABLE]

If $\{\xi_{n}\}\in c_{0}$, then $\xi_{n}^{*}\downarrow 0$; in this case there exists a bijection $\pi:\mathbb{N}\rightarrow\mathbb{N}$ such that $|\xi_{\pi(n)}|=\xi_{n}^{*}$, $n\in\mathbb{N}$.

Hardy-Littlewood-Polya partial order in the space $\ell_{\infty}$ is defined as follows:

[TABLE]

A non-zero linear subspace $E\subset\ell_{\infty}$ with a Banach norm $\|\cdot\|_{E}$ is called a symmetric (fully symmetric) sequence space if

[TABLE]

Every fully symmetric sequence space is a symmetric sequence space. The converse is not true in general. At the same time, any separable symmetric sequence space is a fully symmetric space.

If $(E,\|\cdot\|_{E})$ is a symmetric sequence space and $E_{h}=\{\xi=\{\xi_{n}\}_{n=1}^{\infty}\in E:\xi_{n}\in\mathbb{R}\ \ \forall\ \ n\in\mathbb{N}\}$, where $\mathbb{R}$ is the field of real numbers, then $(E_{h},\|\cdot\|_{E})$ is a Banach lattice with respect to the natural partial order

[TABLE]

and, in addition,

[TABLE]

Examples of fully symmetric sequence spaces are $(\ell_{\infty},\|\cdot\|_{\infty})$,   $(c_{0},\|\cdot\|_{\infty})$ and

[TABLE]

For any symmetric sequence space $(E,\|\cdot\|_{E})$ the following continuous embeddings hold [4, Ch. 2, § 6, Theorem 6.6]:

[TABLE]

Besides, $\|\xi\|_{E}\leq\|\xi\|_{1}$ for all $\xi\in l_{1}$ and $\|\xi\|_{\infty}\leq\|\xi\|_{E}$ for all $\xi\in E$.

If there is $\xi\in E\setminus c_{0}$, then $\xi^{*}\geq\alpha\mathbf{1}$ for some $\alpha>0$, where $\mathbf{1}=\{1,1,...\}$. Consequently, $\mathbf{1}\in E$ and $E=\ell_{\infty}$. Therefore, either $E\subset c_{0}$ or $E=\ell_{\infty}$.

Now, let $(\mathcal{H},(\cdot,\cdot))$ be a complex infinite-dimensional separable Hilbert space, and let $(\mathcal{B}(\mathcal{H}),\|\cdot\|_{\infty})$ be the $C^{*}$-algebra of bounded linear operators in $\mathcal{H}$. Denote by $\mathcal{K}(\mathcal{H})$ (respectively, $\mathcal{F}(\mathcal{H})$) the two-sided ideal of compact (respectively, finite rank) linear operators in $\mathcal{B}(\mathcal{H})$. It is well known that, for any proper two-sided ideal $\mathcal{I}\subset\mathcal{B}(\mathcal{H})$, we have $\mathcal{F}(\mathcal{H})\subset\mathcal{I}\subset\mathcal{K}(\mathcal{H})$ (see, for example, [16, Proposition 2.1]).

Denote $\mathcal{B}_{h}(\mathcal{H})=\{x\in\mathcal{B}(\mathcal{H}):x=x^{\ast}\}$, $\mathcal{B}_{+}(\mathcal{H})=\{x\in\mathcal{B}(\mathcal{H}):x\geq 0\}$, and let $\mathrm{Tr}:\mathcal{B}_{+}(\mathcal{H})\to[0,\infty]$ be the canonical trace on $\mathcal{B}(\mathcal{H})$, that is,

[TABLE]

where $\{\varphi_{j}\}_{j\in J}$ is an orthonormal basis in $\mathcal{H}$ (see, for example, [18, Ch. 7, E. 7.5]).

Let $\mathcal{P}(\mathcal{H})$ be the lattice of projections in $\mathcal{H}$. If $\mathbf{1}$ is the identity of $\mathcal{B}(\mathcal{H})$ and $e\in\mathcal{P}(\mathcal{H})$, we will write $e^{\perp}=\mathbf{1}-e$.

Let $x\in\mathcal{B}(\mathcal{H})$, and let $\{e_{\lambda}\}_{\lambda\geq 0}$ be the spectral family of projections for the absolute value $|x|=(x^{*}x)^{1/2}$ of $x$, that is, $e_{\lambda}=\{|x|\leq\lambda\}$. If $t>0$, then the $t$-th generalized singular number of $x$, or the non-increasing rearrangement of $x$, is defined as

[TABLE]

(see [8]).

A non-zero linear subspace $X\subset\mathcal{B}(\mathcal{H})$ with a Banach norm $\|\cdot\|_{X}$ is called symmetric (fully symmetric) if the conditions

[TABLE]

(respectively,

[TABLE]

imply that $y\in X$ and $\|y\|_{X}\leq\|x\|_{X}$.

The spaces $(\mathcal{B}(\mathcal{H}),\|\cdot\|_{\infty})$ and $(\mathcal{K}(\mathcal{H}),\|\cdot\|_{\infty})$ as well as the classical Banach two-sided ideals

[TABLE]

are examples of fully symmetric spaces.

It should be noted that for every symmetric space $(X,\|\cdot\|_{X})\subset\mathcal{B}(\mathcal{H})$ and all $x\in X$, $a,b\in\mathcal{B}(\mathcal{H})$,

[TABLE]

Remark 2.1**.**

If $X\subset\mathcal{B}(\mathcal{H})$ is a symmetric subspace and there exists a projection $e\in\mathcal{P}(\mathcal{H})\cap X$ such that $\mathrm{Tr}(e)=\infty$, that is, $\dim e(\mathcal{H})=\infty$, then $\mu_{t}(e)=\mu_{t}(\mathbf{1})=1$ for every $t\in(0,\infty)$. Consequently, $\mathbf{1}\in X$ and $X=\mathcal{B}(\mathcal{H})$. If $X\neq\mathcal{B}(\mathcal{H})$ and $x\in X$, then $e_{\lambda}=\{|x|>\lambda\}$ is a finite-dimensional projection, that is, $\dim e_{\lambda}(\mathcal{H})<\infty$ for all $\lambda>0$. This means that $x\in\mathcal{K}(\mathcal{H})$, hence $X\subset\mathcal{K}(\mathcal{H})$. Therefore, either $X=\mathcal{B}(\mathcal{H})$ or $X\subset\mathcal{K}(\mathcal{H})$.

If $x\in\mathcal{K}(\mathcal{H})$, then $|x|=\sum\limits_{n=1}^{m(x)}\mu(n,x)p_{n}$ (if $m(x)=\infty$, the series converges uniformly), where $\{\mu(n,x)\}_{n=1}^{m(x)}$ is the set of singular values of $x$, that is, the set of eigenvalues of the compact operator $|x|$ in the decreasing order, and $p_{n}$ is the projection onto the eigenspace corresponding to $\mu(n,x)$. Consequently, the non-increasing rearrangement $\mu_{t}(x)$ of $x\in\mathcal{K}(\mathcal{H})$ can be identified with the sequence $\{\mu(n,x)\}_{n=1}^{\infty}$, $\mu(n,x)\downarrow 0$ (if $m(x)<\infty$, we set $\mu(n,x)=0$ for all $n>m(x)$).

Let $(X,\|\cdot\|_{X})\subset\mathcal{K}(\mathcal{H})$ be a symmetric space. Fix an orthonormal basis $\{\varphi_{n}\}_{n=1}^{\infty}$ in $\mathcal{H}$, and denote by $p_{n}$ be the projection on the one-dimension linear subspace $\mathbb{C}\cdot\varphi_{n}\subset\mathcal{H}$. It is clear that the set

[TABLE]

(the series converges uniformly), is a symmetric sequence space with respect to the norm $\|\xi\|_{E(X)}=\|x_{\xi}\|_{X}$. Consequently, each symmetric subspace $(X,\|\cdot\|_{X})\subset\mathcal{K}(\mathcal{H})$ uniquely generates a symmetric sequence space $(E(X),\|\cdot\|_{E(X)})\subset c_{0}$. The converse is also true: every symmetric sequence space $(E,\|\cdot\|_{E})\subset c_{0}$ uniquely generates a symmetric space $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})\subset\mathcal{K}(\mathcal{H})$ by the following rule (see, for example, [15, Ch. 3, Section 3.5]):

[TABLE]

In addition,

[TABLE]

We will call the pair $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ a Banach ideal of compact operators (cf. [12, Ch. III], [16, Ch. 1, §1.7]). It is known that $(\mathcal{C}_{p},\|\cdot\|_{p})=(\mathcal{C}_{l_{p}},\|\cdot\|_{\mathcal{C}_{l_{p}}})$ for all $1\leq p<\infty$ and $(\mathcal{K}(\mathcal{H}),\|\cdot\|_{\infty})=(\mathcal{C}_{c_{0}},\|\cdot\|_{\mathcal{C}_{c_{0}}})$. In addition, $\mathcal{C}_{1}\subset\mathcal{C}_{E}\subset\mathcal{K}(\mathcal{H})$  and  $\|x\|_{\mathcal{C}_{E}}\leq\|x\|_{1},\ \ \|y\|_{\infty}\leq\|y\|_{\mathcal{C}_{E}}$ for all $x\in\mathcal{C}_{1},\ y\in\mathcal{C}_{E}.$ Note also that every separable Banach ideal of compact operators is a fully symmetric ideal.

If $(E,\|\cdot\|_{E})$ is a symmetric sequence space (respectively, $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ is a Banach symmetric ideal), then the Köthe dual $E^{\times}$ (respectively, $\mathcal{C}_{E}^{\times}$) is defined as

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

It is known that $(E^{\times},\|\cdot\|_{E^{\times}})$ is a symmetric sequence space [13, Ch. II, §4, Theorems 4.3, 4.9]  and $\ell_{1}^{\times}=\ell_{\infty}$. In addition, if $E\neq\ell_{1}$ then $E^{\times}\subset c_{0}$. Therefore, if $E\neq\ell_{1}$, the space $(\mathcal{C}_{E}^{\times},\|\cdot\|_{\mathcal{C}_{E}^{\times}})$ is a symmetric ideal of compact operators.

A Banach symmetric ideal $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ is said to be perfect if $\mathcal{C}_{E}=\mathcal{C}_{E}^{\times\times}$ (see, for example, [11]). It is clear that $\mathcal{C}_{E}$ is perfect if and only if $E=E^{\times\times}$.

A symmetric sequence space $(E,\|\cdot\|_{E})$ (a Banach symmetric ideal $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$) is said to possess Fatou property if the conditions

[TABLE]

and  $\sup\limits_{k\geq 1}\|\xi_{k}\|_{E}<\infty$  (respectively, $\sup\limits_{k\geq 1}\|x_{k}\|_{\mathcal{C}_{E}}<\infty)$ imply that there exists an element $\xi\in E$  (respectively, $x\in\mathcal{C}_{E})$ such that $\xi_{k}\uparrow\xi$  and  $\|\xi\|_{E}=\sup\limits_{k\geq 1}\|\xi_{k}\|_{E}$  (respectively,  $x_{k}\uparrow x$  and   $\|x\|_{\mathcal{C}_{E}}=\sup\limits_{k\geq 1}\|x_{k}\|_{\mathcal{C}_{E}})$.

It is known that $(E,\|\cdot\|_{E})$ (respectively, $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ has the Fatou property if and only if $E=E^{\times\times}$ [14, Vol. II, Ch. 1, Section a] (respectively, $\mathcal{C}_{E}=\mathcal{C}_{E}^{\times\times}$ [5, Theorem 5.14]). Therefore $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ is a perfect Banach symmetric ideal if and only if $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ has the Fatou property. Note that every perfect Banach symmetric ideal is a fully symmetric ideal.

If $y\in\mathcal{C}_{E}^{\times}$, then a linear functional f_{y}(x)=\mathrm{Tr}\big{(}x\cdot y\big{)},\ x\in\mathcal{C}_{E}, is continuous on $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$, in addition, $\|f_{y}\|_{\mathcal{C}_{E}^{\ast}}=\|y\|_{\mathcal{C}_{E}^{\times}}$, where $(\mathcal{C}_{E}^{\ast},\|\cdot\|_{\mathcal{C}_{E}^{\ast}})$ is the dual of the Banach space $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ (see, for example, [11]). Identifying an element $y\in\mathcal{C}_{E}^{\times}$ and the linear functional $f_{y}$, we may assume that $\mathcal{C}_{E}^{\times}$ is a closed linear subspace in $\mathcal{C}_{E}^{\ast}$. Since $\mathcal{F}(\mathcal{H})\subset\mathcal{C}_{E}^{\times}$, it follows that $\mathcal{C}_{E}^{\times}$ is a total subspace in $\mathcal{C}_{E}^{\ast}$, that is, the conditions $x\in\mathcal{C}_{E},\ f(x)=0$ for all $f\in\mathcal{C}_{E}^{\times}$ imply $x=0$. Thus, the weak topology $\sigma(\mathcal{C}_{E},\mathcal{C}_{E}^{\times})$ is a Hausdorff topology, in addition $\mathcal{F}(\mathcal{H})$ (respectively, $\mathcal{F}(\mathcal{H})^{h}$) is $\sigma(\mathcal{C}_{E},\mathcal{C}_{E}^{\times})$-dense in $\mathcal{C}_{E}$ (respectively, in $\mathcal{C}_{E}^{h}$).

3. Skew-Hermitian operators in $\mathcal{C}_{E}^{h}$

A semi-inner product on a real linear space $X$ is a form $[\cdot,\cdot]\colon X\times X\to\mathbb{R}$ which satisfies

$(i)$. $[\alpha x+y,z]=\alpha\cdot[x,z]+[y,z]$ for all $\alpha\in\mathbb{R}$  and   $x,y,z\in X$;

$(ii)$. $[x,\alpha y]=\alpha\cdot[x,y]$ for all $\alpha\in\mathbb{R}$  and   $x,y\in X$;

$(iii)$. $[x,x]\geq 0$ for all $x\in X$  and  $[x,x]=0$ implies that $x=0$;

$(iv)$. $|[x,y]|^{2}\leq[x,x]\cdot[y,y]$ for all $x,y\in X$.

(see, for example, [6, Ch. 2, §1]).

The function $\|x\|=\sqrt{[x,x]}$ is the norm on a linear space $X$. Conversely, if $(X,\|\cdot\|_{X})$ is a normed real linear space, then there exists semi-inner product $[\cdot,\cdot]$ on $X$ compatible with the norm $\|\cdot\|_{X}$, that is, $\|x\|_{X}=\sqrt{[x,x]}$ for all $x\in X$ [6, Ch. 2, §1]. In particular, the semi-inner product, which is compatible with the norm $\|\cdot\|_{X}$, can be defined using the equation $[x,y]=\varphi_{y}(x)$, where $\varphi_{y}\in X^{*}$, $\|\varphi_{y}\|_{X^{*}}=\|y\|_{X}$ and $\varphi_{y}(y)=\|y\|_{X}^{2}$ (such functional is called a support functional at $y\in X$) ([6, Ch. 2, §1, Theorem 10].

Let $(X,\|\cdot\|_{X})$ be a real Banach space, and let $[\cdot,\cdot]$ be a semi-inner product on $X$ which is compatible with the norm $\|\cdot\|_{X}$. A linear bounded operator $H\colon X\to X$ is said to be skew-Hermitian, if $[H(x),x]=0$ for all $x\in X$ ([10], Ch. 9, §4), in particular, $\varphi_{x}(H(x))=0$ for every $x\in X$.

The following Proposition is well known ([10, Ch. 9, §4, Proposition 9.4.2]).

Proposition 3.1**.**

Let $(X,\|\cdot\|_{X})$ be a real Banach space and let $H\colon X\to X$ be a skew-Hermitian operator. If  $V\colon X\to X$ is a surjective linear isometry then an operator $V\cdot H\cdot V^{-1}$ is a skew-Hermitian.

Let $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ be a separable or perfect Banach symmetric ideal,   $\mathcal{C}_{E}\neq\mathcal{C}_{2}$. Let $H\colon\mathcal{C}_{E}^{h}\to\mathcal{C}_{E}^{h}$ be a skew-Hermitian operator. We want to prove Theorem 1.2, i.e. we will show that there exists $a\in\mathcal{B}(H)^{h}$ such that $H(x)=i(xa-ax)$ for all $x\in\mathcal{C}_{E}^{h}$. To solve this problem, we use a modification of the the original proof of Sourour Theorem 1 [17].

For vectors $\xi,\eta\in\mathcal{H}$, denote by $\xi\otimes\eta$ the rank one operator on $\mathcal{H}$ defined by the equality $(\xi\otimes\eta)(h)=(h,\eta)\xi,\ h\in\mathcal{H}$. It is easily seen

[TABLE]

for any $x\in\mathcal{B}(\mathcal{H})^{h}$ and $\xi,\eta\in\mathcal{H}$. If $y=\xi\otimes\xi,\ \|\xi\|_{\mathcal{H}}=1$, then $y$ is an one dimensional projection on $\mathcal{H}$  and  $\|y\|_{\mathcal{C}_{E}}=\|y\|_{\infty}=1$. Thus for a linear functional

[TABLE]

we have that

[TABLE]

In addition, if $x\in\mathcal{C}_{E}^{h}$ and $\|x\|_{\mathcal{C}_{E}}\leq 1$ then

[TABLE]

Consequently, $\|f_{y}\|_{(\mathcal{C}_{E}^{h})^{*}}=1=\|y\|_{\mathcal{C}_{E}}$. This means that $f_{y}$ is a support functional at $y\in\mathcal{C}_{E}^{h}$, and $[x,y]=f_{y}(x)$ is a semi-inner product on $\mathcal{C}_{E}^{h}$ compatible with the norm $\|\cdot\|_{\mathcal{C}_{E}^{h}}$  ([6, Ch. 2, §1, Theorem 10].

Proposition 3.2**.**

If $\xi,\eta\in\mathcal{H},\ (\eta,\xi)=0$, then $\langle H(\eta\otimes\eta),\xi\otimes\xi\rangle=0$.

Proof.

We can assume that $\|\eta\|_{\mathcal{H}}=\|\xi\|_{\mathcal{H}}=1$. Since  $p=\eta\otimes\eta$ is one dimensional projections and $H$ is a skew-Hermitian operator, it follows that

[TABLE]

By Lemma 9.2.7 ([10, Ch. 9, §9.2], see also the proof of Lemma 11.3.2 [10, Ch. 9, §11.3]), there exists a vector $\xi=\{\xi_{1},\xi_{2}\}\in(\mathbb{R}^{2},\|\cdot\|_{E}),\ \xi_{1}>0,\xi_{2}>0,\ \|\xi\|_{E}=1$, such that the functional  $f(\{\eta_{1},\eta_{2}\})=\eta_{1}\xi_{1}+\eta_{2}\xi_{2},\ \{\eta_{1},\eta_{2}\}\in\mathbb{R}^{2},$ is a support functional at  $\xi$  for space  $(\mathbb{R}^{2},\|\cdot\|_{E})$.

Let us show that the linear functional

[TABLE]

is a support functional at $x$  for  $(\mathcal{C}_{E}^{h},\|\cdot\|_{E})$.

Since $f$ is support functional at $\xi$  for  $(\mathbb{R}^{2},\|\cdot\|_{E})$  and  $\|\xi\|_{E}=1$, it follows that $\xi_{1}^{2}+\xi_{2}^{2}=f(\{\xi_{1},\xi_{2}\})=f(\xi)=\|\xi\|_{E}^{2}=1.$ Furthermore, by $\|f\|=\|\xi\|_{E}=1$, we have that  $|f(\{\eta_{1},\eta_{2}\})|=|\xi_{1}\eta_{1}+\xi_{2}\eta_{2}|\leq 1$  for every $\{\eta_{1},\eta_{2}\}\in\mathbb{R}^{2}$  with $\|\{\eta_{1},\eta_{2}\}\|_{E}\leq 1$.

Further, by Lemma 4.1 [12, Ch. II, §4], we have

[TABLE]

that is, $\{(y(\eta),\eta),(y(\xi),\xi)\}\prec\prec\{\mu(1,y),\mu(2,y)\}$. Since $(E,\|\cdot\|_{E})$ is a fully symmetric sequence space, it follows that

[TABLE]

Consequently, if $y\in\mathcal{C}_{E}^{h}$  and  $\|y\|_{\mathcal{C}_{E}}\leq 1$, then

[TABLE]

that is, $\|\varphi\|_{(\mathcal{C}_{E}^{h},\|\cdot\|_{E})^{*}}\leq 1$. Since $\|x\|_{\mathcal{C}_{E}}=\|\xi\|_{E}=1$ and

[TABLE]

it follows that $\|\varphi\|_{(\mathcal{C}_{E}^{h},\|\cdot\|_{E})^{*}}=1=\|x\|_{\mathcal{C}_{E}}$  and  $\varphi(x)=\|x\|_{\mathcal{C}_{E}}^{2}$. This means that $\varphi$ is a support functional at $x$  for space  $(\mathcal{C}_{E}^{h},\|\cdot\|_{\mathcal{C}_{E}})$.

Hence,

[TABLE]

Since $\langle H(p),p\rangle=\langle H(q),q\rangle=0$ (see (1)), it follows that

[TABLE]

We extend $\eta_{1}=\eta,\ \eta_{2}=\xi$ up to an orthonormal basis $\{\eta_{i}\}_{i=1}^{\infty}$, and let $p_{i}=\eta_{i}\otimes\eta_{i}$. Now we replace our operator $H$ with another skew-Hermitian operator $H_{0}$. Let $u$ be a unitary operator such that $u(\eta_{1})=\eta_{2},u(\eta_{2})=\eta_{1}$ and $u(\eta_{k})=\eta_{k}$ if $k\neq 1,2$. It is clear that $u^{*}=u^{-1}=u,\ up_{1}u=p_{2},\ up_{2}u=p_{1},\ up_{i}u=p_{i},\ i\neq 1,2$, and $V(x)=uxu^{*}=uxu$ is an surjective isometry on $\mathcal{C}_{E}^{h}$, in addition, $V^{-1}=V$.

By Proposition 3.1, a linear operator $H_{1}=VHV^{-1}$ is a skew-Hermitian operator, in particular, $\langle H_{1}(p_{k}),p_{k}\rangle=0$ for all $k\in\mathbb{N}$ (see (1)).

If $i,j\neq 1,2$, then

[TABLE]

[TABLE]

If $i=1,\ j\neq 1,2$, then

[TABLE]

[TABLE]

Similarly, we get the following equalities

$(i)$. $\langle H_{1}(p_{2}),p_{j}\rangle=\langle H(p_{1}),p_{j}\rangle$ if $i=2,\ j\neq 1,2$;

$(ii)$. $\langle H_{1}(p_{i}),p_{1}\rangle=\langle H(p_{i}),p_{2}\rangle$ if $j=1,\ i\neq 1,2$;

$(iii)$. $\langle H_{1}(p_{1}),p_{2}\rangle=\langle H(p_{2}),p_{1}\rangle$ if $i=1,\ j=2$;

$(iv)$. $\langle H_{1}(p_{2}),p_{1}\rangle=\langle H(p_{1}),p_{2}\rangle$ if $i=2,\ j=1$.

It is clear that $H_{0}=\frac{1}{2}(H-H_{1})$ is a skew-Hermitian operator, and if  $i,j\neq 1,2$,  then  $\langle H_{0}(p_{i}),p_{j}\rangle=\frac{1}{2}(\langle H(p_{i}),p_{j}\rangle-\langle H_{1}(p_{i}),p_{j}\rangle)=0$. Similarly, if $i=1,\ j\neq 1,2$ (respectively, $i=2,\ j\neq 1,2$) we get $\langle H_{0}(p_{1}),p_{j}\rangle=\frac{1}{2}(\langle H(p_{1}),p_{j}\rangle-\langle H(p_{2}),p_{j}\rangle)$ (respectively, $\langle H_{0}(p_{2}),p_{j}\rangle=\frac{1}{2}(\langle H(p_{2}),p_{j}\rangle-\langle H(p_{1}),p_{j}\rangle)$), that is, $\langle H_{0}(p_{1}),p_{j}\rangle+\langle H_{0}(p_{2}),p_{j}\rangle=0$ in the case $j\neq 1,2$.

Similarly, $\langle H_{0}(p_{j}),p_{1}\rangle+\langle H_{0}(p_{j}),p_{2}\rangle=0$ if $j\neq 1,2$. Since

[TABLE]

it follows that $\langle H_{0}(p_{1}),p_{2}\rangle=\langle H(p_{1}),p_{2}\rangle$. Similarly, we get that $\langle H_{0}(p_{2}),p_{1}\rangle=-\langle H(p_{1}),p_{2}\rangle$. Finally, since $H_{0}$ is a skew-Hermitian operator, we have $\langle H_{0}(p_{k}),p_{k}\rangle=0$ for all $k\in\mathbb{N}$ (see (1)).

Let $n$ be the smallest natural number such that the norm $\|\cdot\|_{E}$ is not Euclidian on $\mathbb{R}^{n}$. Then there exist (see, [1, Lemma 5.4]) linear independent vectors $\xi=(\xi_{1},\xi_{2},\ldots,\xi_{n}),\ \eta=(\eta_{1},\eta_{2},\ldots,\eta_{n})\in\mathbb{R}^{n},\ \|\xi\|_{E}=1$, such that

[TABLE]

where $f_{\eta}(\zeta)=\sum\limits_{i=1}^{n}\zeta_{i}\eta_{i},\ \zeta=(\zeta_{1},\zeta_{2},\ldots,\zeta_{n})\in\mathbb{R}^{n}$. By rearranging the coordinates we may assume that $\xi_{1}\eta_{2}\neq\xi_{2}\eta_{1}$.

Let $x=\sum\limits_{j=1}^{n}\xi_{j}p_{j}$,  $y=\sum\limits_{j=1}^{n}\eta_{j}p_{j}$, and let $\varphi_{y}(z)=\langle z,y\rangle=\sum\limits_{j=1}^{n}\eta_{j}\cdot\mathrm{Tr}(p_{j}z),\ z\in\mathcal{C}_{E}^{h}$.

Let us show that $\varphi_{y}$  is a support functional at $x$  for  $(\mathcal{C}_{E}^{h},\|\cdot\|_{E})$.  Since $\|f_{\eta}\|_{E^{*}}=1$ (see (3)), it follows that $|f_{\eta}(\zeta)|=|\sum\limits_{i=1}^{n}\eta_{i}\zeta_{i}|\leq 1$ for every $\zeta=\{\zeta_{i}\}_{i=1}^{n}\in\mathbb{R}^{n}$ with  $\|\zeta\|_{E}\leq 1$. Note that $\|x\|_{\mathcal{C}_{E}}=\|\xi\|_{E}=1$.

We should show that $\|\varphi_{y}\|=\|x\|_{\mathcal{C}_{E}}=1$ and $\varphi_{y}(x)=\|x\|_{\mathcal{C}_{E}}^{2}=1$. Indeed, $\varphi_{y}(x)=\langle x,y\rangle=\langle\sum\limits_{j=1}^{n}\xi_{j}p_{j},\sum\limits_{j=1}^{n}\eta_{j}p_{j}\rangle=\sum\limits_{j=1}^{n}\xi_{j}\eta_{j}=f_{\eta}(\xi)=1=\|x\|_{\mathcal{C}_{E}}^{2}$.

If $z\in\mathcal{C}_{E}^{h},\ \|z\|_{\mathcal{C}_{E}}\leq 1$ then $|\varphi_{y}(z)|=|\sum\limits_{j=1}^{n}\eta_{j}(z(\eta_{j}),\eta_{j})|\leq 1.$ The last inequality follows from

[TABLE]

(see Lemma 4.1. [12] Ch. II, §4). Therefore

[TABLE]

This means that $\varphi_{y}$ is a support functional at $x$  for  $(\mathcal{C}_{E}^{h},\|\cdot\|_{E})$.

Consequently,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Let now $\tilde{x}=\xi_{1}p_{1}+\xi_{2}p_{2}-\xi_{3}p_{3}-\ldots-\xi_{n}p_{n}$ and $\tilde{y}=\eta_{1}p_{1}+\eta_{2}p_{2}-\eta_{3}p_{3}-\ldots-\eta_{n}p_{n}$. As above, we have that $\varphi_{\tilde{y}}(\cdot)=\langle\cdot,\tilde{y}\rangle$ is a support functional at $\tilde{x}$. Consequently,

[TABLE]

[TABLE]

[TABLE]

Summing $(4)$ and $(5)$ we obtain

[TABLE]

that is,

[TABLE]

∎

Proposition 3.3**.**

Let $\eta\in\mathcal{H},\ \|\eta\|_{\mathcal{H}}=1,\ p=\eta\otimes\eta,\ x\in\mathcal{K}(\mathcal{H})^{h}$, and let $\mathrm{Tr}(xq)=0$ for any one dimensional projection $q$ with $qp=0$ . Then there exists $f\in\mathcal{H}$ such that $x=\eta\otimes f+f\otimes\eta-(\eta\otimes\eta)(f\otimes\eta),\ \|f\|_{\mathcal{H}}\leq\|x\|_{\infty}$.

Proof.

If $q$ is an one dimensional projection with $qp=0$ then $qxq=\alpha q$ for some $\alpha\in\mathbb{R}$, and $0=\mathrm{Tr}(xq)=\mathrm{Tr}(qxq)=\mathrm{Tr}(\alpha q)=\alpha,$ that is, $\alpha=0$ and $qxq=0$. Let

[TABLE]

If $y\neq 0$ then there exists $r\in\mathcal{P}(\mathcal{H}),\ \dim r(\mathcal{H})=1$ such that $r\leq q+e$ and $rxr=ryr=\beta r$ for some $0\neq\beta\in\mathbb{R}$. Since $rp=0$, it follows that $0=\mathrm{Tr}(xr)=\mathrm{Tr}(rxr)=\beta\neq 0$. Thus $y=0$. Continuing this process, we construct a sequence of finite-dimensional projections $g_{n}\uparrow(I-p)$ such that $g_{n}xg_{n}=0$ for all $n\in\mathbb{N}$, where $I(h)=h,\ h\in\mathcal{H}$. Consequently, $(I-p)x(I-p)=0$.

If $f=x(\eta)$ then $xp=f\otimes\eta$ and $px=\eta\otimes f$. In addition,

[TABLE]

that is, $(I-p)xp=(I-p)f\otimes\eta$. Therefore,

[TABLE]

∎

Proposition 3.4**.**

Let $\eta\in\mathcal{H},\ \|\eta\|_{\mathcal{H}}=1,\ p=\eta\otimes\eta$. Then there exists $f\in\mathcal{H}$ such that

[TABLE]

Proof.

If $x=H(\eta\otimes\eta),\ \xi\in\mathcal{H},\ (\eta,\xi)=0,\ q=\xi\otimes\xi,$  then by Proposition 3.2 we obtain that

[TABLE]

Using Proposition 3.3, we have that there exists $f\in\mathcal{H}$ such that

[TABLE]

Since $H$ is a skew-Hermitian operator, it follows that

[TABLE]

[TABLE]

[TABLE]

Thus $(\eta,f)=0$ and $x=\eta\otimes f+f\otimes\eta-(\eta\otimes\eta)(f\otimes\eta)=\eta\otimes f+f\otimes\eta$. In addition,

[TABLE]

∎

Proposition 3.5**.**

There exists $a\in\mathcal{B}(\mathcal{H})$ such that $H(x)=ax+xa^{*}$ for every $x\in\mathcal{C}_{E}^{h}$.

Proof.

Let $\{p_{i}\}_{i=1}^{\infty}=\{\eta_{i}\otimes\eta_{i}\}_{i=1}^{\infty}$ be a basis in real linear space $\mathcal{F}(\mathcal{H})^{h}$, where $\{\eta_{i}\}_{i=1}^{\infty}$ is an orthonormal basis of $\mathcal{H}$. For every $\eta_{i}\in\mathcal{H}$ there exists $f_{i}\in\mathcal{H}$ such that $H(\eta_{i}\otimes\eta_{i})=\eta_{i}\otimes f_{i}+f_{i}\otimes\eta_{i}$, and $\|f_{i}\|_{\mathcal{H}}\leq\|H\|$ for all $i\in\mathbb{N}$ (see Proposition 3.4). Define a linear operator $a\colon\mathcal{H}\to\mathcal{H}$ setting $a(\eta_{i})=f_{i}$. Since $\|f_{i}\|_{\mathcal{H}}\leq\|H\|$ for all $i\in\mathbb{N}$, it follows that $a\in\mathcal{B}(\mathcal{H})$, in addition, $H(p_{i})=\eta_{i}\otimes a(\eta_{i})+a(\eta_{i})\otimes\eta_{i}$. Since $\eta_{i}\otimes a(\eta_{i})=(\eta_{i}\otimes\eta_{i})a^{*}$ and $a(\eta_{i})\otimes\eta_{i}=a(\eta_{i}\otimes\eta_{i})$, it follows that $H(x)=ax+xa^{*}$ for all $x\in\mathcal{F}(\mathcal{H})^{h}$.

If $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ is a separable space then $\mathcal{F}(H)^{h}$ is dense in $(\mathcal{C}_{E}^{h},\|\cdot\|_{\mathcal{C}_{E}})$. Consequently, $H(x)=ax+xa^{*}$ for all $x\in\mathcal{C}_{E}^{h}$.

Let now $(\mathcal{C}_{E},\|\cdot\|_{\mathcal{C}_{E}})$ be a perfect Banach symmetric ideal. Repeating the proof of Theorem 4.4 [2], we obtain that the skew-Hermitian operator $H$ is a $\sigma(\mathcal{C}_{E}^{h},\mathcal{C}_{E}^{\times})$-continuous. Since the space $\mathcal{F}(\mathcal{H})^{h}$ is $\sigma(\mathcal{C}_{E}^{h},\mathcal{C}_{E}^{\times})$-dense in $\mathcal{C}_{E}^{h}$, it follows that $H(x)=ax+xa^{*}$ for all $x\in\mathcal{C}_{E}^{h}$. ∎

The following Proposition completes the proof of Theorem 1.2.

Proposition 3.6**.**

$a=ib$* for some $b\in\mathcal{B}(\mathcal{H})^{h}$.*

Proof.

If $a=a_{1}+ia_{2},\ a_{1},a_{2}\in\mathcal{B}(\mathcal{H})^{h}$, then

[TABLE]

where $S_{1}(x)=a_{1}x+xa_{1},\ S_{2}(x)=i(a_{2}x-xa_{2}),\ x\in\mathcal{C}_{E}^{h}$. Since $H$ and $S_{2}$ are skew-Hermitian, it follows that $S_{1}=H-S_{2}$ is also skew-Hermitian.

If $p\in\mathcal{P}(\mathcal{H}),\ \dim p(\mathcal{H})=1$, then the lineal functional $\varphi(y)=\langle y,p\rangle=\mathrm{Tr}(yp),\ y\in\mathcal{C}_{E}^{h}$, is support functional at $p$. Thus $\mathrm{Tr}(pa_{1}p+pa_{1})=\mathrm{Tr}(S_{1}(p)p)=0,$ that is, $-\mathrm{Tr}(pa_{1})=\mathrm{Tr}(pa_{1}p)=\mathrm{Tr}(pa_{1}).$ This means that $\mathrm{Tr}(pa_{1})=0$ for all $p\in\mathcal{P}(\mathcal{H})$ with $\dim p(\mathcal{H})=1$. Consequently, $\mathrm{Tr}(xa_{1})=0$ for all $x\in\mathcal{F}(\mathcal{H})$, and by [3, Lemma 2.1] we have $a_{1}=0$. Therefore, $a=ia_{2}$. ∎

4. Skew-Hermitian operators in Orlicz, Lorentz, and Marcinkiewicz ideals of compact operstors

In this section we present applications of Theorem 1.2 to Orlicz, Lorentz and Marcinkiewicz ideals of compact operators.

1. Let $\Phi$ be an Orlicz function, that is, $\Phi:[0,\infty)\to[0,\infty)$ is left-continuous, convex, increasing and such that $\Phi(0)=0$ and $\Phi(u)>0$ for some $u\neq 0$ (see, for example, [7, Ch. 2, § 2.1], [14, Ch. 4]). Let

[TABLE]

be the corresponding Orlicz sequence space, and let

[TABLE]

be the Luxemburg norm in $l_{\Phi}$. It is well-known that $(l_{\Phi},\|\cdot\|_{\Phi})$ is a symmetric sequence space and the norm $\|\cdot\|_{\Phi}$ has the Fatou property.

If $\Phi(u)>0$ for all $u\neq 0$, then $\sum\limits_{n=1}^{\infty}\Phi\left(\frac{1}{a}\right)=\infty$ for each $a>0$, hence $\mathbf{1}=\{1,1,...\}\notin l_{\Phi}$ and $l_{\Phi}\subset c_{0}$. Consequently, we can define Orlicz ideal of compact operators

[TABLE]

Now, Theorem 1.2 yield the following.

Theorem 4.1**.**

Let $\Phi$ be an Orlicz function such that $\Phi(u)>0$ for all $u\neq 0$ and $l_{\Phi}\neq l_{2}$. Let $H\colon\mathcal{C}_{\Phi}^{h}\to\mathcal{C}_{\Phi}^{h}$ be a skew-Hermitian operator. Then there exists self-adjoint operator $a\in\mathcal{B}(\mathcal{H})$ such that $H(x)=i(xa-ax)$ for all $x\in\mathcal{C}_{\Phi}^{h}$.

2. Let $\psi$ be a concave function on $[0,\infty)$ with $\psi(0)=0$ and $\psi(t)>0$ for all $t>0$, and let

[TABLE]

the corresponding Lorentz sequence space. The pair $(\Lambda_{\psi},\|\cdot\|_{\psi})$ is a symmetric sequence space and the norm $\|\cdot\|_{\psi}$ has the Fatou property (see, for example, [13, Ch. II, § 5]). Besides, if $\psi(\infty)=\infty$, then $\mathbf{1}\notin\Lambda_{\psi}$ and $\Lambda_{\psi}\subset c_{0}$. Consequently, we can define Lorentz ideal of compact operators

[TABLE]

Therefore, in the case $\varphi(\infty)=\infty$, Theorem 1.2 is valid for any Lorentz ideal $\mathcal{C}_{\psi}$ of compact operators.

3. Let $\psi$ be as above, and let

[TABLE]

the corresponding Marcinkiewicz sequence space. The space $(M_{\psi},\|\cdot\|_{M_{\psi}})$ is a symmetric sequence space and the norm $\|\cdot\|_{M_{\psi}}$ has the Fatou property (see, for example, [13, Ch. II, § 5]). In addition, $\mathbf{1}\notin M_{\psi}(\mathbb{N})$ if and only if $\lim\limits_{t\to\infty}\frac{\psi(t)}{t}=0$ [13, Ch. II, § 5]). Consequently, in this case, $M_{\psi}\subset c_{0}$ and we can define Marcinkiewicz ideal of compact operators

[TABLE]

Therefore, in the case $\lim\limits_{t\to\infty}\frac{\psi(t)}{t}=0$, Theorem 1.2 is valid for any Marcinkiewicz ideal $\mathcal{C}_{M_{\psi}}$ of compact operators.