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)=Sni=1 AiXBi are known as elementary operators. For these operators we may define the generalized adjoint by E*(X)= Sni=1 Ai*XBi*, and we say that E is normal if EE*=E*E. (in the case when E:C2® C2, 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 dT, defined by dT (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)=Sni=1 AiXBi, where Ai and Bi 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 En-1 is orthogonal to the kernel of the operator E.
J. Anderson (1973) showed that equality B(H)= Cl Im(dN ) Å ker dN 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)=Sni=1 AiXBi –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(EA,B) Å ker EA,B ?
Problem 4. When is B(H)=Cl Im(E'A,B) Å ker E'A,B ?