**Comments and some open problems**

**Open Problems useful for researchers and Postgraduate students. **

**The sum of the range and the kernel of elementary operators**

Operators of the form *E(X)=**S*^{n}_{i=1} A_{i}XB_{i }are known as elementary operators. For these operators we may define the generalized adjoint by *E*(X*)=* **S*^{n}_{i=1} A_{i}*XB_{i}*, and we say that *E* is normal if *EE*=E*E*. (in the case when *E:C*_{2}*® C*_{2}, this definition coincides with the usual one). It is easy to see that *TT*=T*T* ia a necessary and sufficient condition for the mapping d_{T}, defined by d_{T} *(X)=TX-XT*, to be normal. Also, *AC=CA,BD=DB* together with *AA*=A*A, BB*=B*B, CC*=C*C,* *DD*=D*D*, ensures that the mapping *E(X)=AXB+CXD* is normal. In general, elementary operators *E(X)=**S*^{n}_{i=1} A_{i}XB_{i}, where *A*_{i}_{ }and *B*_{i} are commuting families of normal operators are called normally represented elementary operators, and it is easy to see that every normally represented elementary operator is normal.

The following open problems naturally arise.

**Problem**** 1**. Is the condition that *E * is normally represented necessary and sufficient for its normality ?

**Problem 2**. We conjecture that the range of the operator E^{n-1} is orthogonal to the kernel of the operator E.

J. Anderson (1973) showed that equality B(H)= *Cl Im(**d*_{N} *) *Å *ker **d*_{N} is true only in some very special cases, namely when the spectrum of *N* is finite. It would be interesting to find the condition which ensures the validity of *I* =*Cl Im(E)* Å *ker E*, when *E* is the elementary operator defined by *E'(X)=AXB+CXD* or by *E(X)= E(X)=**S*^{n}_{i=1} A_{i}XB_{i} –*X* considered as operators mapping *I* into *I*, where *I* is an ideal of compact operators and *Cl Im(E)* is the closure of the range of *E *in the norm topology.

**Problem 3.** When is *B(H)=**Cl Im*(*E*_{A,B}) * *Å *ker E*_{A,B }?

**Problem 4.** When is * **B(H)=**Cl Im*(*E'*_{A,B}) * *Å *ker E'*_{A,B }?