On the Fine Spectrum of the Operator Defined by the Lambda Matrix over the Spaces of Null and Convergent Sequences

Table 1

Subdivision of spectrum of a linear operator.

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