Suppose that H = l²,  (e_n) is the standard orthonormal basis for H and (l_n) is a dense sequence in K. Set T(å_{n = 1}^¥ a_ne_n) = å_{n = 1}^¥l_n a_n e_n where (a_n) Î l². Obviously K Í sp(T). If l Ï K, then inf{|l- m|; m Î K } > 0 and so S(å_{n = 1}^¥a_n e_n) = å_{n = 1}^¥(l- l_n)^-1a_n e_n is a well-defined operator on H. S is the inverse of lI - T. Therefore l Ï sp(T). Thus K = sp(T).
For every n,   Te_n = l_n e_n. In fact {l₁, l₂, ¼} is the set of all eigenvalues of T that is dense in sp(T).