Find alinear operatort:c[0,1] to c[0,1] whose spectrum is a given interval
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Functional Analysis With Applications", written by Kreyzig and I got the trouble with problem 1 and problem 2 in section 7.3, which are:
Problem 1:
"Let X=C[0;1]X=C[0;1] and define T:X→XT:X→X by Tx=vxTx=vx, where v∈Xv∈X is fixed. Find σ(T)σ(T). Note that σ(T)σ(T) is closed. "
Problem 2:
"Find a linear operator T:C[0;1]→C[0;1]T:C[0;1]→C[0;1]whose spectrum is a given interval [a,b][a,b]."
There is a hint for problem 1 that σ(T)σ(T) is the range of vv. However, I have no idea of proving this.
Could you please help me to explain those
Problem 1:
"Let X=C[0;1]X=C[0;1] and define T:X→XT:X→X by Tx=vxTx=vx, where v∈Xv∈X is fixed. Find σ(T)σ(T). Note that σ(T)σ(T) is closed. "
Problem 2:
"Find a linear operator T:C[0;1]→C[0;1]T:C[0;1]→C[0;1]whose spectrum is a given interval [a,b][a,b]."
There is a hint for problem 1 that σ(T)σ(T) is the range of vv. However, I have no idea of proving this.
Could you please help me to explain those
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