Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space ℓ_𝑝, (1<𝑝<∞)

Table 1

Subdivisions of spectrum of a linear operator.

		1	2	3
		𝑇𝜆^−1 exists and is bounded	𝑇𝜆^−1 exists and is unbounded	𝑇𝜆^−1 does not exist
𝐴	𝑅(𝜆𝐼−𝑇)=𝑋	𝜆∈𝜌(𝑇,𝑋)	—	𝜆∈𝜎𝑝(𝑇,𝑋)
				𝜆∈𝜎ap(𝑇,𝑋)
			𝜆∈𝜎𝑐(𝑇,𝑋)	𝜆∈𝜎𝑝(𝑇,𝑋)
𝐵	𝑅(𝜆𝐼−𝑇)=𝑋	𝜆∈𝜌(𝑇,𝑋)	𝜆∈𝜎ap(𝑇,𝑋)	𝜆∈𝜎ap(𝑇,𝑋)
			𝜆∈𝜎𝛿(𝑇,𝑋)	𝜆∈𝜎𝛿(𝑇,𝑋)
		𝜆∈𝜎𝑟(𝑇,𝑋)	𝜆∈𝜎𝑟(𝑇,𝑋)	𝜆∈𝜎𝑝(𝑇,𝑋)
𝐶	𝑅(𝜆𝐼−𝑇)≠𝑋	𝜆∈𝜎𝛿(𝑇,𝑋)	𝜆∈𝜎ap(𝑇,𝑋)	𝜆∈𝜎ap(𝑇,𝑋)
			𝜆∈𝜎𝛿(𝑇,𝑋)	𝜆∈𝜎𝛿(𝑇,𝑋)
		𝜆∈𝜎co(𝑇,𝑋)	𝜆∈𝜎co(𝑇,𝑋)	𝜆∈𝜎co(𝑇,𝑋)