Hi Clifford, Kimberly here from landscape solutions for you. Thank you for asking the question on your Zoysia grass.
It is recommended to only dethatch every three years. I would recommend in the dead areas to loosen the soil, (tiller or shovel) and add good quality top soil. You live in zone seven, and a good mixture would be sand, black dirt, and saw dust. This mixture will work well. It is common to have problems with grass dying around trees. Bring top soil in and apply about four to six inches of the top soil. You can put down some Zoysia seed, and also put down some hay to help maintain the moisture in the ground. Water on a regular schedule and you should see the seed plugs start to develop.
You can also fertilize in May, July, or August.
I live in zone seven also, and with our weather conditions having been in drought and lately we have received rain, our soil is changing.
I do recommend soil testing, and I leave that up to my clients. I would go ahead and loosen the dead area, apply top soil, seed, and water.
Thank you again for asking the question on your Zoysia and let us know what you decide to do and how things turn out. Feel free to visit anytime here at landscape solutions for you and ask questions. Have a great day!
Kimberly